Burgess went over to put the visitors ahead and the boot of George Williams ensured they led 8-4 at the break despite Vincent Duport's try.
After the restart Duport scored in the corner as Catalans took a 10-8 lead.
But Burgess put Wigan back ahead and further tries by Williams and Sam Tomkins followed by another Burgess effort saw Wigan home.
The Warriors move up to seventh, four points clear of ninth-placed Warrington with two games remaining before the Super 8s. And they are now four points behind fourth-placed Hull - and the top four spot that would give them the chance to defend their title.
Catalans are 10th, four points behind Huddersfield, who are eighth.
A Wigan win or any sorts looked in doubt when Duport collected a last-tackle kick to go over, with Luke Walsh adding the two points to give the French side their two-point advantage.
But Wigan, who were without coach Shaun Wane as he recovers after a hip operation, then dominated.
A brilliant team try ended with Burgess going over in the corner and Williams skipped through to give them a 10-point lead.
A Tony Gigot error then gifted Tomkins the away side's fourth try and the Burgess treble came when he went over from close range.
Catalans head coach Steve McNamara: "For 60 to 65 minutes, there was not a lot to choose between the two teams - in fact I think we were two tries to one up.
"It was really pleasing that we were in that contest but really disappointed with the last 15 minutes. I don't think we deserved that scoreline.
"Having said that, if you give Wigan those opportunities they'll take them and that's what they did. I think that our performance today was a whole lot better than our win over Leigh last weekend.
"We can take a lot of positives out of that and hopefully we can improve next week as much as we have this week going into next week's game against St Helens."
Wigan assistant John Winder: "I was really pleased with the result. For 60 minutes I thought it was a real arm wrestle and it was a really tough game.
"Both sides were very competitive. We just stuck at it and got through that 60-minute mark and things just started to open up for us.
"I thought we got what we deserved and we stuck with the process long enough and we managed to snag a couple of tries before the momentum went with us at the end.
"It was very hot out there and a really physical game. The quality of the game was really high, the opening 15 minutes saw both teams complete in the high eighties and there weren't a lot of errors."
Catalans: Gigot, Broughton, Inu, Wiliame, Duport, Walsh, Albert, Moa, Aiton, Casty, Anderson, Horo, Baitieri.
Substitutes: Bousquet, Da Costa, Margalet, Belmas.
Wigan: S. Tomkins, Davies, Gelling, Gildart, Burgess, Williams, Leuluai, Nuuausala, McIlorum, Sutton, Bateman, Farrell, O'Loughlin.
Substitutes: Isa, Clubb, Powell, Tautai.
Joe Burgess scored a hat-trick as Wigan produced an emphatic final 20 minutes to win at Catalans Dragons.