The 38-year-old Djokovic dropped the first set but recovered to win 6-7 (5/7), 6-3, 6-3, 6-1 and stay in the hunt for an unprecedented 25th Grand Slam singles title.
“I wasn’t really happy about my tennis for the first part of the match but also credit to Zach for playing some high quality tennis,” said Djokovic.
The Serbian star admitted to feeling “not that great” after a laboured performance but is optimistic he will gradually find his rhythm.
“That’s what I’m hoping, the deeper I go into the tournament the better I feel about my game,” he said. “Obviously a bit different for me the last couple of years body-wise. I get the wear and tear quicker than I used to do.”
Djokovic, a four-time champion in New York, made a sluggish start against his 145th-ranked opponent who had won just two main-draw matches at majors.
The 22-year-old Svajda snatched a tight first set in a tie-break when he smacked a bruising backhand down the line that Djokovic could not return.
Seventh seed Djokovic broke for a 4-2 lead in the second set on the way to levelling up the match in bright sunshine on Arthur Ashe Stadium.
Djokovic dropped serve early in the third set but then rattled off five consecutive games to grab the set after falling 3-1 behind.
With Svajda wilting and receiving treatment to both legs, Djokovic pulled away to victory by winning 11 of the final 12 games to register a record-tying 191st hard-court match win at Grand Slams.
He also surpassed the mark he shared with Roger Federer by making the last 32 of a major for the 75th time. He will next play Britain’s Cam Norrie, who advanced after a four-set thriller against Francisco Comesana of Argentina. (AFP)