Melbourne's heatwave had relented by the time the 31-year-old Nadal walked on Margaret Court Arena, but the Spaniard was on fire as he battered the 28th seed into submission.
The sole blemish was a dropped service game early in the second set but it only briefly slowed his charge.
"I am in the fourth round. That's because I am doing the right things. Let's see how far I can go," Nadal, beaten in the final by Roger Federer last year, told reporters.
Nadal arrived in Australia with question marks over the state of his knees after he was forced to withdraw mid-tournament from the ATP Finals in November.
Yet back in the kind of sleeveless top in which he launched his spectacular assault on the grand slams in 2005, and wielding his trusty forehand like a lethal weapon, doubts about Nadal's title prospects have quickly been cast aside.
"I needed hours on court, hours of practice, hours of playing sets with different players," Nadal said of his build-up to the year's first grand slam.
"I think I did a good preparation. I feel I'm playing well."
Argentina's 24th seeded Diego Schwartzman, who Nadal plays next, might prove a tougher obstacle than Dzumhur.
The 25-year-old from Sarajevo had actually walked off court a winner against Nadal in 2016 when the Mallorcan retired from their match at the Miami Masters with an illness.
This time it was Dzumhur who might have sought a quick way out after being allowed two points in the first four games as Nadal started like the scorching wind that had blown through Melbourne Park earlier in the day.
Nadal was 5-0 up in no time before Dzumhur managed to hold serve, prompting a sympathetic cheer from the crowd.
Another flurry of forehand winners helped Nadal into a 2-0 lead in the second set but Dzumhur, who became the first Bosnian to win a Tour level title last year, broke back with a dipping backhand that was too good for Nadal.
Normal service was quickly resumed though and Nadal hammered home a backhand winner to end a superb 26-stroke rally on his way to breaking again in the next game.
Nadal scrapped his way through a 10-minute game at the start of the third and rolled on to victory, sealing it with an ace before belting a ball high into the night sky.
(Reporting by Martyn Herman; Editing by Toby Davis)
