The World Cup holders were indebted to Maxwell and Bailey for their fourth-wicket partnership of 137 and then a late onslaught from Wade as they looked for the win that would give them victory in the five-match series.
After winning the toss, Australia, 2-1 up after losing by 93 runs at Old Trafford on Tuesday, were in trouble at 30-3 as David Willey took early wickets for England.
But Maxwell and Bailey laid the foundations for a testing total in good batting conditions before Wade and John Hastings made up for a middle order-collapse with an unbeaten stand of 81 off just 54 balls as England were set 300 to win.
The left-arm seamer removed opener Joe Burns, bowled off an inside edge as he attempted to leave, before trapping Australia captain Steven Smith leg before wicket with a full-pitch inswinger.
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Aaron Finch was next to go as he edged behind to county team-mate Jonny Bairstow before Maxwell and Bailey repaired the damage.
Maxwell survived two dropped catches on six and 35 but counter-attacked superbly as he made a 12th one-day international 50 off 42 balls with seven fours and a six.
Bailey went on to make his 18th one-day fifty off 64 balls with five fours and guided the tourists past 200 before becoming one of two players to fall with the score on 210 off the bowling of Liam Plunkett in the 41st over as England applied the brakes.
Debutant Marcus Stoinis followed shortly after but some late hitting from Wade and Hastings restored Australia's momentum.
Hastings, called up from county cricket at Durham to supplement Australia's injury-hit squad in his first ODI appearance since 2011, contributed 34 off 26 balls with three fours and two sixes.