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 before the threshold was met for a suspension of play that lasted three hours.
On the resumption, 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.
The 32-year-old Argentinian is ranked 91st and has never won a main-draw match in Melbourne so on paper this was a kind draw but Evans made a poor start, dropping serve in his opening game.
He fought back well with a run of four games in a row but Bagnis was proving a dogged competitor and, although Evans recovered from 1-4 in the second set, a volley dumped into the tramlines allowed the Argentinian to level.
It was nip and tuck in the third before Evans gained the advantage with a break for 4-3, and he took the set before organisers announced the heat stress scale had hit five, the mark where continuing play outside is deemed dangerous.
Play finally resumed at 5pm, and Evans sealed the win after three hours and 29 minutes to set up a second-round clash with veteran Frenchman Jeremy Chardy.