Spinner Swann took five for 28 as the tourists were dismissed for 176 at Chester-le-Street, despite a half-century from captain Ricky Ponting.
England still managed to make a meal of chasing a modest total, slumping from 106 without loss to 141 for five before Paul Collingwood and Tim Bresnan dragged them to an inglorious four-wicket victory.
Swann blamed England's lacklustre performances in the one-day series on the matches being played in the aftermath of their 2-1 Ashes triumph.
"It has been the classic 'after the Lord Mayor's show', it really has felt like that," Swann said.
"It's good that we managed to grab a win at the end, 7-0 would have been embarrassing, it's bad enough at 6-1 but 7-0 would have been a world record and a horrible world record to hold. We're happy we've avoided that."
Skipper Andrew Strauss (47) and Joe Denly (53) then gave England a solid platform with a 106-run opening partnership, prior to the significant wobble among the middle order.
Although the win ensured England escaped an unprecedented series drubbing, another below-par batting performance – Strauss and Denly aside – gave them little cause for optimism ahead of flying to South Africa for the ICC Champions Trophy.
Strauss said: "We need to be honest and not settle for mediocrity.
"Anyone who has been part of this series does not want to go through this experience again."