Maxwell, who earlier took three wickets in two overs, was man of the match with an unbeaten 103 as the Australians chased down England's 155-9 in the 19th over.
Maxwell, who was controversially dropped from Australia's ODI team for the recent series against England, responded with a 58-ball century including 10 fours and four sixes at Bellerive Oval.
"It was nice to get one over the rope and finish the job," said Maxwell. "For us to start the series the way we have has been outstanding. We haven't had a lot of success in T20I cricket for the last few years. Missing out on the one-day squad and not playing any part in the Ashes, it was obviously where I wanted to be this summer. I had to do a lot of work behind the scenes and I feel I've done that and got back into the Aussie colours."
Maxwell had a big moment on 59 when his lofted drive found the fingertips of long-off fielder Jason Roy.
But Maxwell stood his ground and although umpire Gerald Abood's soft signal was out, the television umpire deemed Roy did not complete the catch.
Australia had a rocky start to their run chase when David Willey dismissed David Warner and Chris Lynn in the space of three balls. But Maxwell rebuilt the innings with stands with D'Arcy Short (30), Travis Head (6) and Alex Carey (5 not out) to secure victory.
Willey finished with three wickets for 28 off three overs.
England, troubled by Australia's spinners, lost their last six wickets for 33 in registering a below-par total of 155.
Dawid Malan cracked 50 off 36 balls with five fours and two sixes but England fell away after reaching 94 for two in the 10th over.
Maxwell captured three for 10 and Ashton Agar took two for 15 as Australia sent England's middle order tumbling.
Kane Richardson (1-27) began the damage by removing dangerous opener Jason Roy for nine, before Malan hit his lively half-century.
He was supported by Alex Hales with 22 from 15 balls, and captain Eoin Morgan, also 22 from 14 balls.
Hales was the first of two caught-and-bowled wickets for Agar. When Morgan went, caught by Warner off Maxwell, it sparked a middle-order collapse.
Jos Buttler (5) fell cheaply, caught by Maxwell off Marcus Stoinis (1-16), while Sam Billings (10) popped a return catch to Agar.
Malan was caught in the deep by Andrew Tye for Maxwell's second wicket and David Willey (3) was dismissed next ball to give Alex Carey his first international stumping.
Maxwell's hat-trick ball was wide of the leg-stump line.
Adil Rashid (1) got the toe end of the bat on a shot off paceman Billy Stanlake and was well caught by Stoinis, while Tom Curran (6) was well held by a back-pedalling Warner after failing to pick Tye's knuckle ball.
Chris Jordan hit the final ball for six to finish with 16 from 11 balls, with Mark Wood on five not out. Tye finished with 1-28.
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