The world No 1 extended his winning streak in Grand Slams to 17 matches and took another step towards a sixth Wimbledon crown, the third leg of the calendar-year Slam, and a record-equalling 20th major title.
Djokovic can even still emulate Steffi Graf in 1988 when she won all four Grand Slams and gold at the Olympics – a feat yet to be repeated.
Not that this was an entirely straightforward win. The 34-year-old had described his second-round victory over Kevin Anderson as “flawless”, but he did not remotely hit those heights against American qualifier Kudla, a Tour journeyman ranked 114.
Instead there was plenty of irritation; double-faults here, a long forehand there – even a couple of service breaks and a third-set tie-break to negotiate.
The unforced error count was also up significantly on the six he registered against Anderson – this time he made 28.
Kudla stayed with the Serbian for the first nine games behind a solid service game, but Djokovic struck in the 10th to sneak the first set.
He won four games in a row across the end of the first and the start of the second, and although Kudla managed to break back Djokovic quickly regained his advantage.
But Kudla began feeding off the crowd inside the Court One outpost Djokovic had been downgraded to in favour of Brits Dan Evans and Andy Murray.
He gave them a glimmer of a comeback when handed a break after Djokovic double-faulted and went 3-0 up in the third.
But at 4-2 Kudla’s unforced error count increased at the wrong time and Djokovic pulled level.
Almost unheard of back-to-back double faults from Djokovic gifted Kudla a 3-0 lead in the tie-break.
But some trademark resolute Djokovic defence, and some poor shot choices from Kudla, saw the reigning champion through 6-4 6-3 7-6 (7).
Djokovic said: “Kudla fought well. He played well, I thought, from the beginning. Until the last game of the first set I thought we were pretty even.
“I won, I think, four games in a row, something like that, started playing better. I mixed it up.
“I had to slow down the pace in the game because he was really handling well the ball, the flat ball, from the back of the court. He has a really good balance. I think his backhand and forehand as well, pretty flat. He moves around pretty well on the surface. He’s a really good player.
“I, myself, from the other side, I can’t say that I’m too pleased with the way I performed. I felt like today I was a bit off, especially with serving.
“Usually my serve worked really well in the first two matches, and generally my serve is a weapon. It wasn’t today.
“But it is what it is. I don’t want to talk about negatives. I just won in straight sets. Honestly, I think I can play better. I hopefully will do in the next round.”