Carew was ruthless in front of goal when he pounced on Onyszko's handling error for Villa's first.
But mid-way through the first half the Danes came to life and a fortuitous goal, Sidwell diverting Atle Roar Haland's cross into his own net on his competitive debut, boosted the home side's confidence.
Bjorn Runstrom, Henrik Hansen, Johan Absalonsen and Hans Henrik Andreasen all had chances for Odense.
They were made to pay for their profligacy when Villa regained the lead, Laursen leaping above the defence to head home Young's floating free-kick.
Laursen went close moments later before Villa were caught out by the long ball and Baye Djiby Fall teed up Christensen.
Aston Villa boss Martin O'Neill told the club's website:
"I'd expect by this time next week we'd be much fitter and that's important.
"It was so important for us to stay in the competition. We've scored two goals and we'll be better for it next week.
"I felt for a little period in the second half we started to look tired, but then we started to force it again and we had a couple of decent opportunities and got the goal to go back in front.
"It was just disappointing that we conceded so late in the game. But it's all set up now for Villa Park next week."
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