The Florida-based Pegula made a business-like job of wrapping up the first set before Bucsa had really settled.
She closed it as the evening shadows lengthened on Court One in 22 minutes, with a hefty serve that the Spaniard could only dump into the net.
Pegula then raced to a 4-0 lead in the second, getting the better of the baseline rallies with pounding groundstrokes.
But Bucsa found some form mid-set, producing deft drop shots and fine angles to claw back four games as Pegula's intensity waned.
Bucsa then saved two match points in the eighth game of the set but could not fight off another in the 10th and Pegula won the match in just over an hour with a searing forehand winner into the corner.
Pegula is hoping to better her third-round finish here last year. She has made it to five other Grand Slam quarter-finals, most recently at the Australian Open last January.
"I definitely feel the best I've felt in my career on grass, which I guess is a little late at 29 but I'll take it," she said. "I definitely feel I can go deep here ... a short-term goal to get into the second week."
Pegula will meet Italy's Elisabetta Cocciaretto (22) in the third round.
