The sixth-seeded American, who struggled with injuries earlier in the year, produced her best form of the tournament so far, winning 89% of her first serve points, cracking 13 winners and breaking serve four times.
Pegula crushed a serve Bouzas Maneiro could not put back into play on match point to advance in 70 minutes on Arthur Ashe Stadium court.
"I just played solid," Pegula said in an on-court interview.
"I didn't do anything crazy. I didn't do anything bad or that amazing. I think I was just very consistent.
"Even when she got back into that second set, I was able to find some energy, get a hold and close it out pretty quick, so I'm proud of that."
The unseeded Bouzas Maneiro struggled to match Pegula's level in the contest and was hampered by six double faults and 29 unforced errors.
Pegula said she was forced make some quick adjustments against an opponent she had not seen before.
"You just have no idea how their ball is, how it comes off their racket, how it's going to be playing on this court, especially with the roof closed," she said.
"We didn't warm up with the roof closed either, so there are a lot of variables that you aren't so sure about. At the beginning I was trying to feel it out and I was able to get some rhythm."
Pegula missed this year's French Open to recover from injuries and after suffering an early exit at Wimbledon, she triumphed at this month's Canadian Open.
The Buffalo, New York native has said she would relish winning her first Grand Slam on home soil and opted not to compete in doubles at Flushing Meadows this year to focus on her singles game.
She has flashed an improved serve and speedy defence through her first three matches and will next face either Diana Shnaider.