Teichmann produced an astonishing fightback to beat fourth seed Svitolina 2-6 6-4 7-6 (7-5) at the Caja Magica.
Unseeded Swiss Teichmann came from a set and a break down to claim the scalp of the world number five, who squandered a commanding 5-1 lead in the final set.
Svitolina saw six match points come and go in an incredible deciding set, opportunities she was left to rue when the 40-ranked Teichmann won a tie-break to end a contest that took two hours and 33 minutes to settle.
World number one Barty extended her winning run on clay to 12 matches with a 6-2 6-1 success over Shelby Rogers.
Barty claimed her third title of the year in Stuttgart last weekend and the Australian needed just an hour to send Rogers packing, losing just two points on her first serve and not facing a single break point.
French Open champion Iga Swiatek was also an emphatic winner, easing past another American in the form of Alison Riske 6-1 6-1.
Swiatek, making her debut in this tournament, has won 16 consecutive matches on clay after blowing Riske away.
Defending champion Kiki Bertens beat 15-year-old Victoria Jimenez Kasintseva 6-4 6-0, while Petra Kvitova progressed when opponent Marie Bouzkova retired due to hand injury after the Czech won the first set.
Garbine Muguruza was forced to withdraw ahead of her meeting with Sloane Stephens due to a leg injury. Belinda Bencic, Angelique Kerber and Johanna Konta were among the other winners in the Spanish capital.
Sabalenka, ranked sixth in the world, put paid to Shelby Rogers 7-6 (8-6) 6-0, as the favourite in the Netherlands reached her second singles final this year.
World number 30 Alexandrova had little difficulty against Veronika Kudermetova as she triumphed 6-3 6-1 to reach her first singles final of the season.
Sabalenka and Alexandrova have split four previous meetings with two wins apiece, with the latter taking the most recent clash in straight sets at the Moscow quarter-finals last year.
Seventh seed Haddad Maia stunned favourite Maria Sakkari in the quarter-finals to tee up the last-four clash with Tereza Martincova in Nottingham.
The Brazilian raced out the blocks once more to progress past Martincova, with the former 6-3 4-1 up before the Czech retired with injury.
Rain stopped play in the second semi-final between Viktorija Golubic and Riske with the scores level at one set each.
Sixth seed Riske capitalised from that point onwards to edge past Golubic 6-3 4-6 6-3.
On Sunday, Garcia moved through to the quarter-finals by beating Alison Riske-Amritraj 6-4 6-1 on Louis Armstrong Stadium.
She has won four matches without dropping a set in the first week of the grand slam in Queens, New York, but that is just the continuation of a sensational hot streak.
Garcia, now 28, has won 30 of her last 34 matches, landing titles in Bad Homburg, Warsaw and Cincinnati along the way.
At the 2011 French Open, ATP superstar Andy Murray was so taken by the 17-year-old Garcia's performance against Maria Sharapova that he predicted: "The girl Sharapova is playing is going to be number one in the world one day."
Perhaps Murray will turn out to be right after all, with that forecast having long hung over Garcia, as well intended as it was at the time.
She reached a peak of number four in 2018 but was down at 79th on the WTA list in late May of this year.
Now she sits 17th and will keep climbing after reaching the last eight at the US Open for the first time.
On a 12-match winning streak at present, Garcia said after sinking the hopes of American Riske-Amritraj: "I'm so excited to be in the quarters of the US Open. It's been a great couple of weeks for me."
She came through qualifying to win the Western and Southern Open in Cincinnati, beating Petra Kvitova in that final after felling three seeds along the way.
Recent wins over Iga Swiatek in Poland and Emma Raducanu at Wimbledon, beating home favourites, have showed Garcia is suddenly fearless.
"I'm really trying to play aggressive, go for my shots, even when I'm tight or even when I don't feel it," she said.
"It's how I improved so much in the last couple of months and I really enjoy playing like that, moving forward, and I'm having fun."
Garcia was not at her most fluent in the first set of Sunday's fourth-round match but improved and finished the contest having hit 30 winners.
She has only played one grand slam quarter-final before, losing to Karolina Pliskova at that stage in the 2017 French Open.
Suddenly, though, she is playing top-five standard tennis again, and Garcia will take some stopping.
"I want to enjoy every single win I have," Garcia said in an on-court interview.
"When you don't have them, you miss them, so I want to enjoy this one and recover and get ready for the big match in the quarter.
"I'm really having fun here in the US. I got a lot of confidence from Cincinnati, really enjoy the good energy in New York."
