Machac, 23, made the brighter start to the match, imposing himself with a flurry of crunching forehands and taking Draper to deuce in the Briton’s opening service game.
But Draper withstood the early barrage, settled in with three aces in his next service game and soon began to take control.
He was gifted the first break when Machac put a routine overhead into the net and then went long, but took full advantage, hammering a scorching backhand down the line to bring up two set points before converting a third.
After showing his power from the baseline in the first, Draper showed his deft touch to start the second with a pair of fine drop shots.
But while there was a swagger to an increasingly confident Draper, his opponent was growing ever more frustrated and heated discussions with coach Daniel Vacek between – and during – games became the norm.
Double faults were becoming an issue for the Czech world number 39 and two in a game enabled his opponent to break at the first time of asking in the second set.
Having got the advantage, Draper was ruthless as he moved serenely around the court, coolly staving off a break-back point before earning his second break to go up 5-1.
Two eye-catching winners helped him serve it out and the prospect of a Machac comeback looked unlikely when the Briton broke in the first game of the third set.
Of the six break points Draper faced five came in his final two service games as Machac – by this stage two breaks down – tried to delay the inevitable but again the British number one kept his cool.
“After the tournament, I can look back and be proud of whatever I have achieved but for now I’m focused on the next round and hopefully going further,” he said.
“One of my main goals was to consistently be great in practice and keep my body robust and just all around get stronger and stronger and I think some of that work’s paying off.
“I’m just glad to be in the position I’m in.”
With a quarter-final place confirmed, another win would see Draper become the first Briton to make the men’s semi-finals in New York since Murray claimed his first Slam in 2012.
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