Sinner, who became embroiled in controversy in the week before the year's final Grand Slam after the International Tennis Integrity Agency said that he tested positive twice for an anabolic agent in March, was slow out of the starting blocks.
The Australian Open champion committed several unforced errors in the opening set, which McDonald took full advantage of to claim an early lead.
The American broke early in the second set to take a 1-0 lead, but after spurning three breakpoints in a lengthy game, Sinner levelled at 1-1.
The hard-fought break appeared to fill the world number one with confidence and the tide of the contest turned as he raced to a 4-1 advantage, before holding serve and breaking once again to square the match at a set apiece.
Sinner never looked back from there and won five games in a row to take the third before winning the first three games of the fourth set without dropping a point.
McDonald won a couple of games late in the match to delay the inevitable, but Sinner made his advantage count and served out the set to claim an emphatic victory.
The 23-year-old faces another American, Alex Michelsen, in the second round.