The World No.4 picked up right where she left off last week in New Haven, where she picked up a 19th career title at the Connecticut Open, and powered to a 6-1, 6-1 victory in just 55 minutes.
"The most important thing that I have quick, nice first round," Radwanska said after the match. "Just very happy to have this quick match just to, you know, be ready and fresh for the next one."
22-year-old Pegula, who reached the second round here last year, was overwhelmed by Radwanska's variety from the start and struggled to find her footing early on. Radwanska won the first twelve points of the match and held a 3-0 lead before Pugula won a point.
The American's nerves wreaked havoc on her big, flat-hitting game, with most of Pegula's 34 unforced errors coming from a groundstroke buried into the net. By contrast, Radwanska hit just four unforced errors the whole match - none in the first set - and won 74 percent of points behind her first serve.
Radwanska will go on to play Britain's Naomi Broady in the second round.
Also in action on a busy Day 2, No.11 seed Carla Suárez Navarro also eased through to the second round, notching a 6-0, 6-0 victory over Teliana Pereira. Katerina Siniakova also got back to her winning ways against Eugenie Bouchard, toppling the Canadian 6-3, 3-6, 6-2.
No.16 seed and 2011 champion Samantha Stosur had a more difficult match against Camila Giorgi, 7-5, 6-7(4), 6-1. The Australian was two points away ending the match in straight sets at 7-5, 5-4, but Giorgi held on to force a third.