"I'm pretty happy with my performance," First matches of the season are always rusty. I was happy that in the important moments I was really composed," Swiatek was quoted as saying by the WTA Tour website.
Swiatek had 22 winners in the clash to Pegula's 14, and she finished with only three more unforced errors than the American.
The top-ranked Swiatek was tested in the match but she stepped up during crucial moments to equal her best performance at the season-ending Grand Slam. The Pole had also reached the fourth round last year.
The 21-year-old, free-swinging Swiatek, a two-time Grand Slam champion, is showing signs that she might be prepared to reach such heights once more.
If Swiatek wins in Cincinnati, she will become the first player since Williams in 2013 to lift seven titles before the US Open.
Gauff, who has never been ranked higher than her current spot of 11th, will overtake Great Britain's Emma Raducanu and likely Russian Daria Kasatkina to move up to ninth when the next rankings are released after prevailing in an exceptionally tight match against her Belarussian opponent
While Swiatek will aim for her second Grand Slam title of the year after clinching the French Open title as part of an incredible 37-match winning streak, Serena, who has won 23 Grand Slam singles titles in her career.
The victory puts the 32-year-old Cornet into the Round of 16 at Wimbledon for the first time since 2014.
French Open champion Swiatek extended her streak but her latest win came after an extended struggle. Poland's Swiatek overcame Dutch lucky-loser Lesley Pattinama Kerkhove 6-4, 4-6, 6-3 in the second round of Wimbledon on Thursday.
Swiatek, the newly-crowned Roland Garros champion, will look to extend her 35-match winning streak when she takes on qualifier Jana Fett at the All England Club.
