Sinner, who has seen two of his biggest rivals for the title at Flushing Meadows fall by the wayside this week after Carlos Alcaraz and Novak Djokovic were beaten, took an hour and 53 minutes to bundle out his 87th-ranked opponent.
The Italian made slow starts in his opening two matches at the US Open but he shot out of the blocks on Arthur Ashe Stadium court for a 5-0 lead before taking the first set on serve.
Australian O'Connell provided a degree of resistance in the second set, but Sinner broke early to take a 2-1 lead and went on to clinch the set with a vicious cross-court forehand winner.
Sinner continued to pepper O'Connell with winners, taking his tally up to 46 as he broke twice in the final set to wrap up the win.
Australian Open champion Sinner's strongest weapon in the victory was his flawless serve, firing 15 aces and denying O'Connell any break points across the three sets.
O'Connell, who had lost all eight of his previous meetings with top-10 players, looked powerless throughout as his bid to reach the fourth round of a Grand Slam for the first time failed.
Sinner faces either 14th-seeded local hope Tommy Paul or Canadian qualifier Gabriel Diallo in the fourth round.
