Poland’s Swiatek needed just over two hours to see off Dutch lucky loser Lesley Pattinama Kerkhove 6-4, 4-6, 6-3.
Swiatek was in early trouble before winning four games in a row to take the first set but went down a break in the second and could not recover.
In the end, though, Swiatek took charge in the decider, breaking for a 3-1 lead and closing out the match.
She surpasses Monica Seles’ 36-match winning streak from 1990 and matches Martina Hingis’s 37-match winning run from the beginning of the 1997 season.
“I would say the grass is pretty tricky for me, I’m not going to lie,” said Swiatek. “I guess you can see that I’m not playing maybe as efficiently as on other surfaces.
“Basically my confidence is getting better overall. But this tournament is tricky and I’m still feeling out how to play the best game here.”
British wildcard Katie Boulter shocked 2021 finalist Karolina Pliskova 3-6, 7-6 (7/4), 6-4 before dedicating her win to her late grandmother, who died this week.
“I’m probably going to get emotional,” she said. “My grandmother died two days ago so I’d like to dedicate that to her today.”
Women’s fourth seed Paula Badosa set up a clash against two-time champion Petra Kvitova after both had straightforward wins.
Former champion Simona Halep beat Kirsten Flipkens 7-5, 6-4 on Thursday to advance to the third round.
Both players struggled to hold serve throughout their match on Court Two, with former world number one Halep breaking her Belgian opponent six times.
The Romanian, who won the title in 2019, has slipped to 18th in the world rankings.
Defeat marks the end of 36-year-old Flipkens’s singles career.
The players embraced at the net and Flipkens, a semi-finalist in 2013, kissed the court.
Halep will play Poland’s Magdalena Frech in the last 32.
