Australia, who won the toss, lost four wickets for 29 runs to be 193 for six in the 37th over.
But wicket-keeper Wade and all-rounder Mitchell Marsh (40 not out), neither of whom played in the World Cup final, turned the tide.
Their unbroken stand of 112 in 13 overs saw Australia to 305 for six after leg-spinner Adil Rashid had taken four for 59.
Left-hander Wade, who took nine balls to score his first run, faced 50 deliveries in total including 12 fours as he atoned for his part in the run-out of Shane Watson.
"Me and Mitchell just knuckled down and took our innings deep," said Wade, who also held three catches as Australia beat England for the 10th time in 11 ODIs.
England captain Eoin Morgan insisted side should have succeeded in their run-chase.
"We lost it in the 10 or 15 overs in the middle of the chase where Australia bowled pretty well" he said.
Morgan added: "I was pleased with Adil Rashid, he has come on in leaps and bounds. For him to come out and hold his nerve, I was very impressed."
After Australia captain Steven Smith opted to bat, openers Joe Burns (44) and David Warner (59) made a solid start.
Warner had a reprieve on 18 when Chris Woakes failed to hold a hard-hit caught and bowled chance.
England, who had won both the preceding Ashes Test series an Monday's lone Twenty20 against Australia in Cardiff, started to choke off the supply of runs.
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