The defending champion was given a thorough workout by Thompson but always looked in control at the crucial moments as he won 6-3, 7-6, 7-5 to move into the third round.
He joins Roger Federer and Serena Williams as the only players to have reached 350 wins at the Slams, but the 36-year-old has his eyes firmly fixed on higher goals.
Playing with his usual clinical accuracy he eventually subdued Thompson who offered up some thing rare at Wimbledon these days - some serve and volley tennis.
It certainly kept Djokovic honest but the second seeded Serb, bidding to win a record-equalling eighth Wimbledon and fifth in a row, always looked in control.
Thompson, ranked 70, was within two points of levelling the match at the sharp end of the second set but once Djokovic came through the tiebreak with a couple of timely aces thrown in, there was only really likely to be one outcome.
That said, Thompson, given loud backing by the Centre Court crowd desperate to see a contest, refused to slink away.
But serving at 5-6 in the third set the Australian could not keep Djokovic at bay and he netted a volley to end the contest.
"He was a bit unlucky in the second set. He had some chances but yeah, he played a great match and deserves a big round of applause for sure," Djokovic said.
While Djokovic certainly does not need any help as he aims for a 24th Grand Slam title to match the all-time record of Margaret Court, things are certainly falling his way.
He has avoided being snagged by the rain delays and the draw looks inviting.
Up next is either Swiss veteran Stan Wawrinka (38), who he famously lost to in the French Open final eight years ago, or Argentina's 29th seed Tomas Etcheverry (23).
Wawrinka and Etcheverry will play their delayed second-round match on Thursday after it was one of several matches cancelled because of rain delays on Wednesday.