Les Kiss's men are top of Pool 1 courtesy of a 20-13 win over La Rochelle, who host out-of-contention Quins next week.
Wasps led 21-0 and having let Quins back into the match, they secured the bonus point as they moved out of sight once more only to endure an alarming late collapse.
Harlequins' interest in Europe was extinguished long ago - they had lost all four previous matches of this season's group phase - but a team weakened by injury and suspension fought to the end and were rewarded.
Mathematically they can still qualify for the knock-out phase, but they face mission impossible against pool leaders Ulster at the Ricoh Arena next weekend to progress.
Quins rallied with substitute Ross Chisholm and James Lang touching down before the break.
Quins' fightback was interrupted 14 minutes after half-time and once again a line-out proved to be their undoing.
When Wasps centre Brendan Macken evaded the defensive efforts of Jamie Roberts to add a bonus-point try and extend the advantage to 16 points, Dai Young would have expected his side to see the tie off.
Christian Wade performed a try-saving tackle on Tim Visser but the pressure continued to build through a succession of short-range drives that were completed when Care burrowed over.
Teenaged fly-half Marcus Smith all but ended Wasps' Champions Cup hopes as he arrived as a second-half replacement to inspire an under-strength Harlequins to an astonishing turnaround victory at Twickenham Stoop.
Haskell was then dismissed by Romain Poite for taking out Jamie Roberts with a shoulder charge off the ball, and Quins' comeback was completed when flanker James Chisholm barged over to seal a remarkable win.
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