Third seed Bencic needed just 88 minutes to see off her 36th-ranked opponent as she made it 15 victories from her past 18 matches.
Czech wildcard Tereza Martincova is also through to the last eight on home soil after battling past French Open finalist Anastasia Pavlyuchenkova 7-6 (7-5) 7-6 (8-6).
Martincova is into her fifth quarter-final of the season and will take on number four seed Maria Sakkari, who saw off Jelena Ostapenko 6-4 6-2 earlier on Thursday.
Also through to round three is Elena Rybakina following a hard-fought 6-3 2-6 6-1 win over qualifier Magda Linette, while Jil Teichmann – who shocked former world number one Angelique Kerber in the previous round – advanced in three sets against Alison Riske.
She will face Petra Kvitova for a place in the semi-finals.
Bouchard took the first set, but Bencic came from behind to advance with a 6-7 (7-9) 6-1 6-4 victory on Tuesday.
Eighth seed Jelena Ostapenko, runner-up in the Korea Open, suffered a premature exit as Aliaksandra Sasnovich consigned the former French Open champion to a 6-2 6-2 defeat.
Emma Raducanu also went out in straight sets, with Daria Kasatkina beating the 2021 US Open champion 7-5 6-4.
Catherine McNally blitzed to a 6-1 6-2 victory over Anna Blinkova, while Alycia Parks progressed past Karolina Pliskova to secure the first top-20 win of her career.
Top seed Ons Jabeur swept aside Ann Li with relative ease in a 6-2 6-3 victory at the Jasmin Open to reach the last 16.
Veronika Kudermetova, the second seed, breezed past Varvara Gracheva with a commanding 6-1 6-0 win.
Anastasia Potapova, Petra Martic, Magdalena Frech and Harriet Dart were among the other victors.
Teenager Gauff, who made the main draw after a pair of wins in qualifying, fought back from a break down three times in the opening set and went on to triumph 7-5 6-4.
A meeting with third seed Aryna Sabalenka is the next task for the American, who has won just two of her past seven matches on the WTA Tour.
Elena Rybakina has reached a Tour-leading five finals in 2020 but the eighth seed fell to a 6-2 3-6 6-3 defeat to the measured Daria Kasatkina.
Kasatkina played an intelligent, efficient match and had 15 winners to 14 unforced errors, while Rybakina had 35 to 49.
Veronika Kudermetova's 6-2 6-4 victory over Donna Vekic booked a second-round meeting with Karolina Pliskova, while there were also wins for Jennifer Brady and Ons Jabbeur.
Third seed Kontaveit, who won this competition last year, retired when a set down to Tereza Martincova, while world number seven Maria Sakkari was defeated 5-7 7-5 7-5 by 21-year-old American Alycia Parks in a big upset.
Parks lost the opener but found two crucial breaks of serve when 6-5 up in both the second and third sets to eliminate the fourth seed.
The withdrawal of Belinda Bencica – seeded sixth – with a foot injury put Barbora Krejcikova into the quarter-finals, while Ekaterina Alexandrova will also be in the last eight after overcoming fifth seed Daria Kasatkina 6-1 7-6 (7-1).
The tournament favourites were in much better form at the Jasmin Open in Tunisia, as the top two seeds sealed their places in the quarter-finals with straight-sets wins.
Favourite Ons Jabeur cruised to a 6-1 6-3 victory over Evgeniya Rodina, converting four of her seven break points to safely advance to the next round of her home tournament.
And number two seed Veronika Kudermetova eased past Magdalena Frech 6-3 6-4 to set up a quarter-final meeting with Diane Parry.
The unseeded Estonian, who is ranked 30th in the world, saw off fourth seed Sakkari 6-2 7-5 to claim the trophy in the Czech Republic.
Kontaveit has now won two tournaments in four weeks, having won in Cleveland last month to end a four-year wait since winning her first title in 2017.
This was Sakkari's first final in over two years as well, but Kontaveit cruised past the Greek in just over an hour and a half as she did not drop a set all week.
That flawless level of tennis saw her eliminate Petra Kvitova, Paula Badosa and Belinda Bencic en route to the final, where the 25-year-old faced only one break point as she powered to a comfortable victory over a player who reached semi-finals at the French Open and US Open this year.
Sakkari managed more of a fight in the second set, where she battled for her sole break point, but Kontaveit breezed past any pressure and kept her serve to maintain control.
Sakkari, who is likely to move into the top 10 despite losing, had boasted a 5-3 head-to-head record in main-draw clashes between the pair coming into Sunday's final, but 23 unforced errors outnumbered the 20 winners from her racket as Kontaveit collected her most notable title to date.
Krejcikova lifted the Tallinn Open last week for her first title of the season, and made it two from two with an impressive run to the final before a hard-fought triumph over Swiatek to earn a ninth straight victory and another championship trophy.
Swiatek herself had won 10 matches on the bounce, and the Pole raced to five of the first six games to lead the opener 5-1, though Krejcikova rattled off four straight games to level it up.
However, after making it 6-5, Swiatek then found a crucial break to clinch the first set.
The second went all the way to a tie-break, which Krejcikova took the ascendancy of by putting herself 6-1 up, and despite a late Swiatek rally, the Czech 26-year-old finally levelled the match after watching four set points come and go.
In the decider, neither could find a break until the eighth game, with Krejcikova doing so to put herself within a game of winning.
And in an extraordinary final stand, Swiatek survived five championship points before her resistance was ultimately broken to deny her an eighth final victory in 2022.
It was Krejcikova's first win against Swiatek, having lost their previous two meetings, as the seventh seed sealed an incredible victory.
World number six Pliskova, already frustrated by second-round exits from the US Open and French Open since tennis returned from lockdown, suffered a 4-6 6-4 6-3 defeat to Russian Veronika Kudermetova, despite leading by a set and a break at one point.
Pliskova also lost to the world number 47 at the Western and Southern Open in August, and this was another blow to add to the list of frustrating results for the Czech in the past two months.
Qualifier Gauff succumbed 1-6 7-5 7-6 (7-2) in her clash with Belarusian Aryna Sabalenka, despite having led by a double break at 5-2 in the deciding set.
The American 16-year-old believes she can use the setback as part of her tennis education, saying she needed to play a "smarter" game.
"I just need to play a little bit smarter on pressure points, to make my opponent play," Gauff said, quoted on the WTA website.
"I feel like I wasn't nervous today - I think I just need to be smarter. Sometimes I have tendencies, when it gets to the end, to rush out of the points when instead I should just put the ball in the court."
Belgian Elise Mertens eased to a 6-4 6-2 win over Czech player Karolina Muchova, while Ons Jabeur beat Jelena Ostapenko 6-4 6-4 and US Open semi-finalist Jennifer Brady came from 5-2 behind in the first set to earn a 7-5 6-2 victory over Russian Daria Kasatkina.
Rybakina, who has struggled for consistency since victory at Wimbledon earlier this year, came back from a set down to dispatch Madison Keys with a 5-7 6-3 6-3 win.
She could face eighth seed Jelena Ostapenko in the next round, but one name not in the draw will be Azarenka, who could not claw back victory against Ekaterina Alexandrova in a 6-4 4-6 2-6 defeat.
Alexandrova is in line to play either Emma Raducanu or Daria Kasatkina in the next stage.
Meanwhile, top seeds Iga Swiatek and Paula Badosa earned the identity of their opponents after receiving first-round byes.
Swiatek will play Ajla Tomljanovic, who progressed in straight sets against Zhang Shuai, as Badosa faces Petra Kvitova, who was made to work a little harder in three by Bernarda Pera.
At the Jasmin Open, third, fifth and seventh seeds Alize Cornet, Elise Mertens and Katerina Siniakova each advanced with respective straight-sets routs of Laura Pigossi, Jaqueline Cristian and Chloe Paquet – the latter retiring as she trailed 4-0 in the second set.
But eighth seed Magda Linette is out after letting an early lead slip against Lucrezia Stefanini, who celebrated a 2-6 6-4 6-4 victory.
Sakkari beat world number five Svitolina 6-3 6-3 to progress to the last eight of a WTA Premier event for the third time this season.
Unseeded Greek Sakkari struck 32 winners and returned superbly, winning all but eight of 29 points on Svitolina's second serve.
The 23rd-ranked Sakkari will face either wildcard Jelana Ostapenko or Ons Jabeur for a place in the semi-finals.
Ostapenko knocked out fifth seed Petra Martic, the 2017 French Open champion easing to a 6-3 6-1 victory.
Martic had won their previous two meetings, but the Croatian was beaten in just 57 minutes at the indoor hard-court event.
Anett Kontaveit was another seed to fall, Sara Sorribes Tormo reaching a maiden Premier-level quarter-final at the expense of the Estonian with a 6-1 6-4 success.
Victoria Azarenka is also two wins away from the final, coming from a set down to see off Barbora Krejcikova 2-6 6-2 6-1, and will now face Karolina Muchova or Elise Mertens, who got past Amanda Anisimova in straight sets.
Swiatek continued her stellar season with a 7-6 (7-5) 2-6 6-4 victory over the Russian.
A tie-break was needed to settle the opening set, with Swiatek securing a crucial break of serve when 6-5 up.
The Pole was caught off guard in the second set as Alexandrova took it the distance, before Swiatek claimed the solitary break of the decider to lead 4-3 and did not allow her opponent a way back.
Swiatek, who has 60 wins to her name in 2022, will do battle with Krejcikova in the championship match after she came from a set down to defeat Wimbledon champion Elena Rybakina 3-6 7-6 (7-4) 6-4.
Rybakina served 17 aces to Krejciokva's two, but the Czech 26-year-old battled to victory to put herself in the final of her home tournament.
At the Jasmin Open, Alize Cornet knocked out second seed Veronika Kudermetova with a 6-4 6-3 triumph to clinch her place in Sunday's final.
Cornet saved nine of 11 break points to beat the world number 12 and has not dropped a set all week at the tournament in Tunisia.
Elise Mertens awaits Cornet in the final after a 6-4 6-0 success over Claire Liu, who knocked out top seed Ons Jabeur on Friday.
Mertens broke Liu at 5-4 up to take the first set before emphatically sealing victory with a sublime second in which she won 24 of the 29 points.
