The defending champion sparked fears that he may have to pull out of the tournament after delaying a practice session, having needed extensive treatment on a shoulder problem during his second-round match.
But Djokovic showed no obvious discomfort in a 6-3 6-4 6-2 win over American Kudla.
"What pleased me most is that I managed to play almost pain free, that's a big improvement on my last match," he said.
"I'm not going to go into medical details, but it was bothering me over the last couple of weeks and especially in the last match."
Djokovic will face former champion Stan Wawrinka in a marquee last-16 match on Sunday.
He added: "We've had some great battles over the years, everywhere.
"I lost to him last time we played here, in the final in 2016. He has a lot of power, a lot of quality. Let the better player win."
Wawrinka got through in straight sets against Italian lucky loser Paolo Lorenzi, the Swiss, seeded 23rd, winning 6-4 7-6 (9) 7-6 (4).
Roger Federer roared past Britain's Dan Evans and will face David Goffin in round four.
Federer, five times a winner at Flushing Meadows, hit a total of 48 winners past the British number two.
Federer wrapped up a 6-2 6-2 6-1 win in just one hour and 20 minutes.
Goffin, the Belgian 15th seed, beat Spain's Pablo Carreno Busta in three tight sets, 7-6 (5) 7-6 (9) 7-5.
Kei Nishikori, the runner-up in 2014, was a third-round casualty.
The Japanese seventh seed was beaten 6-2 6-4 2-6 6-3 by Alex De Minaur of Australia.