The Polish world No. 3 defeated the 81st-ranked American 6-1, 2-6, 6-3 in two hours and one minute at the All England Club, Polish state news agency PAP reported.
Świątek will next face former Wimbledon finalist Karolína Plíšková of the Czech Republic in the second round.
A potential third-round opponent is seven-time Wimbledon champion Serena Williams, who has returned to competition at the age of 44 after receiving a wild card into the tournament.
Williams was due to open her campaign against Australia's Maya Joint later on Tuesday.
Later in the tournament, Świątek could face familiar opponents, including Ukraine's Marta Kostyuk, who beat her at this year's French Open, and American Emma Navarro, who eliminated the Pole at last week's Bad Homburg Open.
Second-seeded Elena Rybakina of Kazakhstan is also in Świątek's half of the draw.
Świątek, 25, is the last Polish woman remaining in the singles after Maja Chwalińska, Magdalena Fręch and Magda Linette were all eliminated in the first round on Monday.
A tearful Iga Świątek celebrates after defeating Taylor Townsend in a hard-fought first-round match at Wimbledon on Tuesday. Photo: EPA/TOLGA AKMEN
No woman has successfully defended the Wimbledon singles title since Serena Williams in 2016.
On the men's side, Poland's Kamil Majchrzak advanced to the second round on Tuesday with a 6-3, 7-5, 7-5 victory over Chile's Alejandro Tabilo.
Fellow Pole Hubert Hurkacz also reached the second round after defeating 11th-seeded Casper Ruud of Norway in straight sets on Monday.
Hurkacz will face Austria's Sebastian Ofner on Wednesday.
Wimbledon, the only grass-court Grand Slam of the year, began on Monday and runs until July 12, when the men's final will be held.
The women's final is scheduled for July 11.
Świątek claimed her first Wimbledon title last year with a 6-0, 6-0 victory over American Amanda Anisimova in the championship match.
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Source: IAR, PAP