The Czech player, ranked 42 in the world, clawed her way back from 4-1 down in the deciding set to win 6-4, 2-6, 6-4.
Pegula was quickly out of the blocks, breaking the former French Open finalist at her first opportunity but that proved a false dawn as she lost three service games in the first set, landing just 41% of her first serves.
The American found more rhythm on her serve in the second set and cut her error count, surging into a 4-1 lead and sealing the set with her second break.
Pegula was in total charge when she broke again in the fourth game of the decider to lead 3-1 before the match was suspended for the roof to be closed due to rain at the All England Club.
She held serve when play resumed to take a 4-1 lead but the 24-year-old Czech, who had never previously never been beyond the second round at Wimbledon, dug deep to pull level.
Pegula faced a real struggle on her serve in the following game and sprayed a forehand long to hand Vondrousova a vital break and she took full advantage.