Conor O'Shea's men seemed dead and buried as the rejuvenated Wasps, who had only avoided relegation on the final day of last season, stormed into a 40-13 lead after 56 minutes.
Christian Wade did the early damage, scoring two blistering tries and providing another for Tom Varndell before forwards Marco Wentzel and Tim Payne touched down.
But man of the match Nick Evans, on his 100th Premiership appearance, scored 22 points to lead Harlequins on one of the great Premiership comebacks.
The reigning champions, back at the scene of their Premiership final victory over Leicester, scored four tries and 29 unanswered points in the final quarter of a compelling match.
Tom Guest came off the bench to launch the comeback, Evans set up two tries for Mike Brown and then touched down himself before wrapping up the victory with a penalty four minutes from time.
For all the drama on show, there was also a hefty dose of controversy as the introduction of new rules expanding the influence of the television match official (TMO) proved crucial.
Wasps thought they had scored through Varndell on the stroke of half-time but the try was chalked off after the TMO confirmed a forward pass early in the move.
Had this game been last season - or not broadcast live on ESPN - Varndell's try would have stood. The debate over the TMO's influence raged on.
Wasps players lay desolate on the turf at full-time. They had two bonus points but knew that, through some weak defence, they had blown a golden opportunity.
The performance of Dai Young's men for the first hour did at least indicate the horrors of last season do now lie behind them.
Harlequins like to play at pace but they could not live with Wasps' rugby. The dark days of March, April and May when the club were desperate to just grind out any kind of result to secure survival seemed long gone.
It took just three minutes for Wasps to fashion the first opening as Varndell released the electric-paced Wade to round Brown and score.
Robinson converted two penalties to extend the lead before the wing twins of Wade and Varndell combined again.
Hugo Southwell ran onto Elliot Daly's flat pass and released Wade, who again scorched past Brown and over the try-line, only to inexplicably pass to Varndell instead of touching down.
If Wade's decision caused brief palpitations in the Wasps coaching box, the mood will have been eased by Varndell touching down and Robinson converted to open that 20-0 lead.
The champions, who had barely had the ball, responded with a quick tap from Danny Care and Williams collected Robshaw's offload to score under the posts.
But Wasps were relentless. After a third Robinson penalty, Jonathan Poff forced the turnover and Wade produced an exceptional finish to score his second try.
In a moment reminiscent of Jason Robinson's try for the Lions in the first Test against Australia in 2001, Wade stood up his opposite number George Lowe and then scorched past him on the outside.
Evans reduced the arrears with two more penalties before Varndell streaked away on the stroke of half-time for what looked like Wasps' bonus-point try.
But the celebrations were cut short after referee Greg Garner utilised the trial rules and asked TMO Graham Hughes to rule on a suspected forward pass from Robinson early in the move.
The try was duly disallowed and for a while it seemed academic.
Seven minutes into the second half, Varndell pounced on a mistake from Mo Fa'asavalu and then flicked the ball in-field for Wentzel to gallop over from the 22m line.
Wentzel then turned provider, exchanging deft passes with Payne for the loose-head prop to crash over the line from close-range at the end of another dynamic attack. Wasps led 40-13.
With the game seemingly gone, Harlequins exploited some weak Wasps tackling to stage their late rally. Initially it seemed a try bonus and maybe a face-saving losing bonus point was the best they could hope for.
Guest came off the bench to score with his first touch before Brown showed some excellent hands to take a quick pass from Evans and dive over in the corner.
When Evans picked out Brown with a raking cross-field kick, the England full-back scored Harlequins' fourth try - but they were not finished yet.
Brown squeezed an offload to Guest and Evans took advantage of more poor defence to wriggle through two tackles and touch down for Harlequins' fifth try.
The conversion took Harlequins to within a point and Evans rounded off a remarkable comeback with his penalty four minutes from time.
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