The British number two, who struggled with illness in the build-up to the tournament, was scheduled first up on what could be the hottest day of the fortnight.
He toiled for nearly three hours to open up a two-sets-to-one lead against his Argentinian opponent before the temperature threshold was met for a suspension of play that lasted from 2pm and 5pm.
On the resumption, 25th seed Evans ensured he was not out on court for longer than was necessary, breaking serve in the seventh game for a 6-4 4-6 6-4 6-4 victory.
The temperature was already above 30 degrees when play began, with Evans looking to repeat the result from the first round in 2017, when he defeated Bagnis in straight sets before making it to the fourth round.
Evans was bothered by the sun on his serve from one end and, asked why he did not wear a cap, said: “I’m pretty good in the heat, doesn’t bother me. I have just never worn a hat and I have never liked wearing one. Doesn’t feel comfortable. Big ears, I guess.
“I was obviously very unlucky. I was first (on the schedule), and that’s where the sun was, and it was super bright. I was messing around with the ball toss. That’s probably the reason I lost the second set.”
Bagnis is ranked 91 and has never won a main-draw match in Melbourne but he made a good fight of the match.
Evans recovered from 2-0 down in the opening set and then fought back from 1-4 in the second only for Bagnis to break again.
He had just made the breakthrough by taking the third set when he was informed the match would be suspended.
“I never even knew about the rule, to be honest,” said Evans. “I came back from the toilet, and then the guy said, ‘You’re going in’. I just knew it was hot. Once the wind stopped, it was horrible.”
Evans would rather have carried on, though, adding: “I had won the third. I felt like he was struggling. He was going to the towel a lot at the back of the court. And then it was quite long, and I had to start again. I didn’t feel great going back out. But then I actually played very well.”
The 32-year-old wrapped up victory after three hours and 29 minutes and will next meet veteran Frenchman Jeremy Chardy, who is playing his first tournament since the US Open in 2021 following knee surgery.