Gauff's often-maligned serve undermined some of her play but the fourth seed refused to become rattled in front of the home fans as she sent over 15 winners. She next plays either fellow American Iva Jovic or the Philippines' Alexandra Eala.
"Today was mainly defence, honestly, in the first set," said 22-year-old Gauff. "Once I got my rhythm I was able to swing through a little more."
Fourth seed Gauff dropped her serve in the third game but was controlling the tempo by the sixth, where she levelled the affair with a series of deep backhand strikes.
"Once I was down that break, I just told myself I need to accelerate through my shots," said Gauff, who showed terrific variety in a 21-shot rally to break Bucsa again in the eighth.
"I'm just happy with how I fought through, even when I wasn't playing my best."
Gauff never let Bucsa get a hold of the momentum in the second set, where she did some of her best work from the baseline and put up fewer unforced errors.
She got the critical break in the ninth game with a terrific backhand winner and was smooth as butter in her pale yellow kit as she tapped home the winning touch from the net.