Edinburgh (3) 16
Try: Robertson Con: Paterson Pens: Paterson 3
Wasps (12) 25
Try: Rees Con: Cipriani Pens: Cipriani 6
Wasps' win in Scotland was vital to their Heineken Cup hopes
A late try from Tom Rees sealed a scrappy Heineken Cup victory for Wasps at Murrayfield.
Danny Cipriani kicked four penalties to give the visitors a healthy advantage but Chris Paterson led the Edinburgh comeback with three successful kicks.
Cipriani struck again before Mark Robertson pounced on a diagonal from Phil Godman to bag a home try.
However, Edinburgh let the slender lead slip, with Cipriani slotting another penalty and Rees plunging over.
Cipriani arrived in the Scottish capital in arguably the worst form of his fledgling career, having been dropped by England after error-strewn displays in the opening two November Tests.
But he kicked 20 points in the Pool Two clash in a near-flawless display with the boot to keep his side's quarter-final hopes alive.
However, a European record Edinburgh crowd of 7,711 must have been disappointed to see the home team thwarted by fussy refereeing from Frenchman Christophe Berdos, who who sent flanker Ross Rennie to the sin bin for a dangerous tackle soon after they had gone in front.
Cipriani slotted over from Rennie's indiscretion to put the visitors back in front and, a few minutes later, Godman's long-range penalty attempt fell just short.
The Edinburgh fly-half had given away the evening's first points when his kick was charged down and his team-mates infringed at the breakdown. Cipriani then flighted over a beautiful 40-yard penalty.
Edinburgh were struggling to cope with Wasps' blitz defence and Cipriani added three more penalties, without the visitors posing any attacking threat.
Paterson cut the deficit from in front of the posts before the interval and added two more kicks thanks to a powerful start to the second half.
Edinburgh were suddenly on top, running and recycling the ball brilliantly, with Nick de Luca bundled into touch as he looked to score.
Cipriani claimed his fifth penalty but the Scots continued to look the more menacing side and were rewarded when Godman's crossfield kick was claimed by Robertson in the right corner.
Paterson made no mistake with the conversion to complete a remarkable turnaround.
But with almost a quarter of the game remaining, Cipriani put Wasps 18-16 ahead with a penalty as Rennie trudged off.
Wasps wrapped up victory three minutes from time when Rees bundled over - with the hint of a double movement - and Cipriani added the extras.
Edinburgh coach Andy Robinson:
"Tactically, we were smart and our pack was in total control of Wasps and we played in the right areas.
"We made a couple of disappointing decisions. Every time we got isolated in our half, Wasps punished us.
"The closest they got to our line with the ball in hand was our 22m line and they got 12 points in the lead. Unfortunately, at this level, that's how you win games.
"When it came to closing the game down the last 15 minutes, Wasps showed the class they had in order to do that."
Wasps rugby director Ian McGeechan:
"You're never sure with the referees what you're going to do with the ball.
"Rightly or wrongly, it's safer to try to play the game in your opponents' half.
"It's not pretty, it's nowhere near the quality of rugby that was being played last year, but there was a lot at stake today. We've just got to keep winning. We couldn't afford to lose tonight."
McGeechan on his returning England internationals:
"Credit to all the England boys, they've had a six-day turnaround.
"When you've played four tough weeks internationally, it was a big ask of them and they've been excellent."
Edinburgh: Paterson; Robertson, Cairns, De Luca, Webster; Godman, M Blair; Jacobsen, Ford, G Cross, C Hamilton, J. Hamilton, MacDonald, Rennie, Hogg.
Replacements: Kelly for Ford (19), Kerr for G. Cross (75), Mustchin for C. Hamilton (59).
Not Used: Newlands, Laidlaw, D Blair, Southwell.
Wasps: Van Gisbergen; Sackey, Waldouck, Flutey, Lewsey; Cipriani, Reddan; Payne, Webber, Vickery, Skivington, Birkett, Betsen, Rees, Haskell.
Replacements: Shaw for Skivington (54), Hart for Rees (80).
Not Used: Varley, French, Simpson, Walder, Mitchell.
Ref: C Berdos (France).