Collins, seeded 10th, had victory in her grasp after reeling off four straight games in the second set.
Yet she was unable to get over the line at the first six times of asking, initially failing to serve out the win before Teichmann saved five match points.
Collins, though, regained momentum in the tie-break to win 6-3 7-6 (7-2).
World number nine Veronika Kudermetova is next up for Collins, having received a walkover to the last eight.
Caroline Garcia, the highest-ranked player left in the tournament, overcame Katerina Siniakova 6-3 3-6 7-5, but was made to work for her win.
Having been pegged back in the second set, Garcia seemed well set to push for victory when she went 3-1 up in the decider, but Siniakova broke and then held her serve to restore parity.
A tie-break looked likely until Garcia converted the third break point of a thrilling 12th game to claim victory.
Barbora Krejcikova, the 2021 French Open champion, was ousted by fifth seed Daria Kasatkina, who cruised through 6-2 7-5.
World number eight Kasatkina will take on two-time Wimbledon champion Petra Kvitova for a place in the last four.
Belinda Bencic also progressed, as did Paula Badosa and Beatriz Haddad Maia.
Anna Blinkova caused an upset at the Hobart International, dispatching top seed Marie Bouzkova 7-5 6-4.
Fifth seed Anhelina Kalinina is the favourite after Bouzkova's exit and a tie with 2019 champion Sofia Kenin is her reward for a win over Tatjana Maria.
Hurkacz made it through a tough encounter against Stan Wawrinka 7-6 (7-5) 6-4 to give Poland a 2-1 lead in their Group B tie with Switzerland in Brisbane, before Linette came from behind to beat Jil Teichmann 5-7 6-4 6-1 to clinch it.
Poland will face Italy in the Brisbane final, whose 5-0 whitewash of Norway included Matteo Berrettini sealing victory with a 6-4 6-4 win over Casper Ruud.
In Perth, Croatia advanced to the City Finals after a dramatic day that saw Caroline Garcia get France on the board with a straight sets win against Petra Martic, before Adrian Mannarino came from a set down against Borna Gojo to serve for the match as he looked to bring his country level at 2-2.
However, Gojo broke back in a tense encounter before winning a deciding tie-break to send Croatia through to face Greece, who sealed a 4-1 win against Belgium.
Stefanos Sakellaridis recovered from going a set behind to beat Zizou Bergs, before Maria Sakkari overcame Elise Mertens 6-1 7-5, with Sakkari and Stefanos Tsitsipas adding some gloss to the score with a mixed doubles success against Kirsten Flipkens and Michael Geerts.
Spain and Australia were unable to progress but the hosts at least had the consolation of securing a victory thanks to wins for Jason Kubler against Albert Ramos-Vinolas, and the mixed doubles duo of John Peers and Sam Stosur against Jessica Bouzas Maneiro and David Vega Hernandez.
The USA had already secured their Finals berth against Great Britain in Sydney, but were ruthless as they finished off their tie against Germany to win 5-0.
After Jessica Pegula beat Laura Siegemund 6-3 6-2, Frances Tiafoe also defeated Oscar Otte in straight sets, before the American duo combined to beat Siegemund and Daniel Altmaier 6-7 (5-7) 6-4 10-7 to complete a clean sweep.
Former world number one Halep was the only player to have won the competition to have reached the last eight, but she was no match for the Tunisian, who won 6-3 6-2 in just over an hour.
After a difficult start to the year, Jabeur has hit her stride in recent times and will be contesting a second semi-final in three tournaments.
While Jabeur caught the eye, particularly with her penchant for a dropshot, Halep proved to be her own worst enemy, recording 12 more unforced errors (20) than winners (eight).
Up next for Jabeur is qualifier Ekaterina Alexandrova, who beat 2019 French Open semi-finalist Amanda Anisimova 6-4 6-3 to reach her first WTA 1000 last-four clash.
Alexandrova has won six of her previous seven meetings with Jabeur.
Meanwhile, Anhelina Kalinina saw her impressive run ended by Jil Teichmann. The Ukrainian had seen off three major winners on the bounce in Sloane Stephens, Garbine Muguruza and Emma Raducanu, but she was beaten in straight sets this time.
Teichmann – who won 6-3 6-4 – will face Jessica Pegula for a place in the final, with the American eliminating Spain's final hope of a home winner in Sara Sorribes Tormo, 6-4 6-2.
Tunisian Jabeur, who beat Olympic gold medallist Belinda Bencic and two-time grand slam winner Simona Halep to reach the semi-finals, needed just an hour and one minute to secure a routine last-four win over Ekaterina Alexandrova.
The world number 10 dominated her Russian opponent in a 6-2 6-3 win, before setting her sights on victory in what will be her sixth career final on the WTA Tour but first at such a high tier.
"I'm going to put a positive here. I want to win this final," Jabeur said after her win. "I'm going to put my heart, my favourite drop shot, my forehand in.
"I'm just going to really give my best. I don't want to regret [anything]. The main important thing for me, I know it's winning the title, but [also] knowing that I gave it all during the match and not regretting that. I know if this one [title] is not coming, then there is another one.
"I keep pushing myself to do better. The proof is that from Charleston [where Jabeur lost the final to Bencic last month], I worked really hard to be in the finals here. Like I said, I'm going to leave my heart on the court on Saturday."
Jabeur will face American Pegula in Saturday's final after she registered a 6-3 6-4 triumph over Switzerland's Jil Teichmann.
The 12th seed was tested when Teichmann fought back from a break down to 4-4 in the second set, with 28-year-old Pegula managing to dig deep to break once more and reach what is also her first WTA 1000 final.
Saturday's contest will represent just the fourth final of Pegula's career, and the first since losing to compatriot Serena Williams in straight sets at the 2020 Auckland Open, but she will enter the top 10 of the WTA rankings with a win.
"I knew I was close to the top 10, but it's so hard, you have to step up and do really well to win a tournament," Pegula said on court.
"I'm just so happy to be in the final, it's my first final in a 1000. I've been knocking on the door in the last few tournaments, [but] I was able to take care of business today."
Jabeur and Pegula have met on four previous occasions with each player boasting two victories each, Jabeur winning their last meeting at the last-16 stage of this year's Dubai Tennis Championships.
The Swedish qualifier, who reached her first WTA final in four years in Merida having had her 2022 derailed by injury, triumphed 7-5 5-7 7-5 over the Ukrainian world number 40 in almost three hours.
Kostyuk, ranked 40th in the world, was not helped by 17 double faults across the match, with Peterson progressing to the second round where she will face 22nd seed Zhang Shuai.
Alize Cornet was also a surprise first-round loser, going down 6-2 7-5 to 427th-ranked Evgeniya Rodina in one hour and 42 minutes.
Rodina's prize for her shock victory is a second-round date with 2023 Australian champion and second seed Aryna Sabalenka.
Camila Giorgi, who beat Peterson for the aforementioned Merida Open crown last month, won through with a routine 6-3 6-3 victory over Arantxa Rus. Giorgi will next face third seed Jessica Pegula.
World number 41 Shelby Rogers was pushed for two hours and 22 minutes by Katie Volynets but triumphed 6-4 4-6 6-1. Rogers will take on seventh seed Maria Sakkari in the second round.
China's Wang Xin, making her Indian Wells debut, defeated former world number 12 Elise Mertens 6-3 6-1, while world number 33 Zhu Lin was downed by Lesia Tsurenko 6-4 6-3.
Aliaksandra Sasnovich won 6-3 7-6 (7-5) over veteran Kaia Kanepi, while 53rd-ranked Linda Fruhvirtova fought back after copping a bagel to win 0-6 6-2 6-3 over 52nd-ranked Mayar Sherif.
Jil Teichman set up a second-round clash with Swiss compatriot Belinda Bencic after a 6-1 6-4 victory over Ashlyn Krueger.
World number 26 Camila Giorgi battled from a break down in both sets to beat fifth seed Muguruza 7-5 6-3 in an hour and 53 minutes.
That meant 12th seed Giorgi made her second straight quarter-final, following a last-eight appearance in Birmingham, and will next face Viktoriya Tomova, who defeated Kirsten Flipkens 3-6 6-3 6-4.
Jil Teichmann, the 10th seed at the tournament, was a notable second-round elimination, falling to a 7-6 (9-7) 4-6 6-3 defeat to Briton Harriet Dart, who later triumphed 6-4 2-6 6-4 over Marta Kostyuk.
Dart's fellow Briton Katie Boulter stunned last year's Wimbledon runner-up Karolina Pliskova in the second round but had no such luck against Petra Kvitova, losing 5-7 6-0 7-5.
Beatriz Haddad Maia, the winner at the Birmingham Open last week, extended to a 12-match winning streak with a 6-1 6-2 victory over Jodie Burrage.
Jelena Ostapenko also progressed after Madison Keys retired when 6-3 down and will next face Anhelina Kalinina, who battled to a 6-3 2-6 6-3 win over 16th seed Yulia Putintseva.
While seeds fell at Eastbourne, there were not as many shocks at the Bad Homburg Open, where Angelique Kerber defeated Lucia Bronzetti 6-2 6-3 to reach the quarter-finals.
Fourth seed Simona Halep also made the last eight with 6-0 6-3 victory over Tamara Zidansek and will meet Amanda Anisimova after she won an All-American match against Ann Li 6-0 6-2.
Alize Cornet downed Tatjana Maria 7-6 (7-4) 6-4 to tee up a clash with Kerber, while Bianca Andreescu will meet top seed Daria Kasatkina after defeating Katie Swan 6-4 6-4.
Four-time grand slam winner Osaka crashed to a resounding 6-3 6-1 loss to Spain's Sara Sorribes Tormo, exiting her first tournament on clay since the 2021 French Open, where she withdrew citing mental health issues.
Osaka, who had posted an underwhelming 20-15 record on the surface prior to this week, looked uncomfortable throughout and appeared to struggle with a leg injury during a disappointing second set display.
The 24-year-old was not the only high profile player to be on the receiving end of a shock during a day of drama in the Spanish capital, as several of the competition's seeds failed to secure places in the last 16.
Another home favourite, Garbine Muguruza, fell to a resounding loss of her own as Anhelina Kalinina raced to a 6-3 6-0 victory over the seventh seed, while sixth seed Danielle Collins was thrashed 6-1 6-1 by Canadian Bianca Andreescu.
Fourth seed Maria Sakkari was the highest-ranked player in action, and although the world number five won the first set of her clash with Daria Kasatkina, the Greek eventually fell to a 3-6 6-3 6-1 loss, while 2021 US Open runner-up Leylah Fernandez went down 6-4 6-4 to Jil Teichmann.
One big name who did make comfortable progress, however, was Fernandez's US Open conqueror Raducanu, who eased to a 6-2 6-1 win over Kostyuk to set up a last-16 encounter with another Ukrainian in Kalinina.
The 19-year-old, who has been quoted as saying she believes clay could prove to be her best surface in the future, was delighted with her victory and enjoying the tournament after dropping just one game in the second set.
"I'm definitely happy with my performance," Raducanu said on court. "Marta's a great opponent - I knew it was going to be a really tough battle. I went out there trying to be really aggressive and it paid off.
"It's my first clay court season and I'm really enjoying it. Madrid is such a cool city and it's got such a great vibe about it. I definitely want to try and stay here for as long as possible."
Potapova won the first set in 39 minutes, claiming the only break of the frame in the sixth game, but Pegula responded by breaking immediately in the second.
Despite squaring the match up, the American trailed 3-1 in the third set, only to fight back again and triumph in two hours and 17 minutes.
Pegula will face 15th seed Petra Kvitova after she won a seesawing three-set contest over 24th seed Jelena Ostapenko, 0-6 6-0 6-4.
Ostapenko won the first six games, before Kvitova won the next 10, only for the Latvian to hit back and claim the next four, squaring up the deciding set at 4-4. But two-time Wimbledon champion Kvitova held her nerve and won the final two games for victory.
World number two Aryna Sabalenka progressed to the final 16 via walkover after her third-round opponent Lesia Tsurenko withdrew.
The 2023 Australian Open champion will take on 2021 French Open winner Barbora Krejcikova after she toppled Wang Xin 6-2 6-7 (1-7) 6-2.
Seventh seed Maria Sakkari secured victory in a two-and-a-half-hour third-round clash with Anhelina Kalinina, winning 3-6 6-2 6-4.
Two-time major runner-up Karolina Pliskova won 6-1 7-5 over Veronika Kudermetova, progressing into the last eight to face Sakkari.
Sixth seed Coco Gauff, who turns 19 on Monday, eased past 54th-ranked fellow teenager Linda Noskova 6-4 6-3 in one hour and 19 minutes.
Sweden's Rebecca Peterson continued her resurgent form with a 3-6 6-3 6-1 win over Jil Teichman, setting up a clash with Gauff.
Ninth-seed Raducanu was the biggest name in action on Tuesday and had been hoping to rack up three successive WTA Tour victories for the first time since her incredible US Open success in September last year.
But Kalinina, who saw off Sloane Stephens and Garbine Muguruza in her two previous matches, got the better of the Brit in an entertaining 6-2 2-6 6-4 win.
Raducanu had not dropped a set in either of her first two outings in Madrid but Kalinina quickly put an end to that.
While the teenager responded well in the second, getting an important break to make it 3-1, Kalinina rallied again in what was a closer deciding set.
The pair traded breaks and then Kalinina got another to set her en route to victory at 5-4 – she had to save one more break point but did ultimately see off Raducanu to book a clash with Jil Teichmann in the last eight.
Teichmann beat Elena Rybakina earlier in the day, dispatching the 16th seed in impressively comfortable fashion as she won 6-3 6-1.
Progression from the last eight for Teichmann will see her reach the semi-final of a WTA 1000 event for the first time.
Sara Sorribes Tormo, the last Spaniard left in the draw, defeated Daria Kasatkina 6-4 1-6 6-3 in something of a rollercoaster ride to line up a quarter-final contest with 12th seed Jessica Pegula.
The American kept her focus despite opponent Bianca Andreescu needing a medical timeout and rain causing a 30-minute halt, eventually winning 7-5 6-1.
Swiatek, who became the first female player to win 25 consecutive matches since Serena Williams in 2015, will face Bianca Andreescu in the last eight after she eased past Petra Martic 6-4 6-4.
"I didn't start well, and everybody could see that," Swiatek told reporters. "I'm really happy with the way I reacted and how I improved in the first set.
"Also how different the second set looked to the first one because I could really reset and really change the way I played. That's the most positive thing for me."
There was a shock in the final game of the day, though, as Daria Kasatkina dumped out number two seed Paula Badosa 6-4 6-4.
Third seed Aryna Sabalenka sealed a third straight win over Jessica Pegula, easing to a 6-1 6-4 victory, while fourth seed Maria Sakkari defeated Coco Gauff 6-4 7-5.
Seventh seed Danielle Collins was knocked out by unseeded compatriot Amanda Anisimova, the 20-year-old cruising past the Australian Open finalist 6-2 6-2.
There were also victories for Jil Teichmann, who overcame Elena Rybakina 6-7 (3-7) 6-3 7-5, and ninth seed Ons Jabeur, who beat Yulia Putintseva 6-3 6-2.
