England Wing Tom Voyce touched down after just nine seconds following an error from Quins centre Will Greenwood, who spilled the kick-off in front of his own posts at the Causeway Stadium.
Somehow Quins hauled themselves back into the game with Greenwood making amends with a try of his own, and 11 points from the boot of Jeremy Staunton.
But their dismal finishing meant they did not deserve to end what is now a 10-game losing streak and Wasps pushed themselves up to second in the table thanks to Voyce's try and conversion and four penalties from Mark Van Gisbergen.
Harlequins went into the game trailing Worcester by four points at the bottom of the table. And when the visitors conceded a record-breaking soft try after a mere nine seconds of the first half, things did not look like improving.
While Van Gisbergen's conversion was sailing over, experts confirmed it was easily the fastest Premiership score, beating Martin Corry's 24-second effort for Leicester against London Irish earlier this season.
It also equalled the previous best in top class rugby, from Scotland's John Leslie against Wales in 1999 with video analysis needed to separate the two.
After that nightmare start, Harlequins held on grimly against a couple of rampaging attacks orchestrated by Matt Dawson and Fraser Waters.
They steadied themselves with two Staunton penalties after 10 and 18 minutes, the second after Wasps lock Simon Shaw was sin-binned for handling in a ruck.
And with a one-man advantage, Greenwood was able to atone for his astonishing error with the try that put his side in front after 24 minutes.
Voyce's miscued clearance kick flew straight to Simon Keogh, who scorched down the left before passing out of the tackle to give Greenwood the simplest of tasks.
Staunton stroked over the conversion from wide on the left for a 13-7 lead Wasps fans could scarcely believe.
Their side responded by adding more structure to a scrappy game and levelled with penalties from Van Gisbergen either side of half-time, the first after Quins captain Andre Vos was the second man to be sin-binned by Ashington referee Dave Pearson.
The quality of the finishing from both sides was woeful in the second half, though.
Wasps were guilty of failing to exploit space out wide when Shaw dithered on the ball, then Harlequins somehow contrived to blow a three-on-one wide on the right.
The odds tilted away from Quins again on the hour – when George Harder was the third man sent to the sin-bin for a ruck offence instantly punished by Van Gisbergen's third penalty.
Then they swung back the other way as Staunton's drop-goal levelled the game again 15 minutes from time, only for Van Gisbergen to nudge his side ahead again five minutes later.
Wasps had to dig in during the last 10 minutes before Greenwood was knocked backwards as he tried to thread his way through a defensive wall to the line.
His side at least picked up a bonus point – their fifth of the season – but still trail Worcester by three having played a game more.
There was some potentially worrying news for England coach Andy Robinson, too, at the interval.
Joe Worsley, a certain starter against Canada next weekend, was substituted after picking up a heavy knock on the thigh.
For London Wasps: Tries: Voyce Con: Van Gisbergen Pens: Van Gisbergen 4
For NEC Harlequins: Try: Greenwood Con: Staunton Pens: Staunton 2 Drop: Staunton