The Warriors led 11-6 approaching the final minute after flanker Sam Betty's second-half try looked to have secured four priceless points.
Wasps had been the Warriors' last Premiership victims on March 1, 2013, since when Worcester had gone 21 league games without a win.
But Wasps prop Jake Cooper-Woolley touched down from close-range, and former Worcester fly-half Andy Goode converted with the game's final kick to edge Wasps home.
Worcester are now 10 points adrift of 11th-placed Newcastle with just five league games left, and they will travel to Tyneside on Sunday week knowing they have to win following an excruciating 13-11 defeat at Wasps' hands.
Two of Worcester's remaining fixtures are away against Premiership title contenders Bath and Saracens, which compounds their degree of difficulty, and it is feasible that they might not recover.
Betty's 53rd-minute touchdown left Wasps floundering, along with penalties from full-back Chris Pennell and fly-half Ryan Lamb, yet Goode came up trumps by adding a conversion to an earlier penalty double, and Worcester were sunk.
Wasps, Amlin Challenge Cup quarter-finalists, are destined for a mid-table Premiership finish, and much of their latest league performance lacked snap and precision despite England lock Joe Launchbury making a quickfire return to domestic action following his mighty displays in the RBS 6 Nations Championship.
While Launchbury delivered a trademark hard-working shift, there was little else for Wasps rugby director David Young to enthuse about, apart from full-back Elliot Daly's assured contribution and Cooper-Woolley's late score.
Worcester boss Dean Ryan will now train his sights towards Kingston Park in nine days' time knowing that victory would kick-start hopes of completing an astonishing escape from the drop, but it appears an unlikely scenario.
Scotland prop Euan Murray returned from Six Nations duty to pack down in Worcester's front-row, while former Warriors loosehead Matt Mullan featured for Wasps alongside ex-Warriors goalkicker Goode.
Worcester, despite their lowly league status, made a bright start and deservedly went ahead when Pennell landed a seventh-minute penalty from just inside the Wasps half.
Wasps could not establish any early momentum, but they also suffered a degree of disruption when wing Josh Bassett went off injured, which meant scrum-half Joe Simpson moving out wide and Charlie Davies filling Simpson's regular number nine role.
Both teams looked rusty after a three-week break from Premiership action, yet Worcester - with Lamb pulling the strings - could take far more from an opening quarter high on endeavour, but low on skill.
Wasps relied on a combination of Daly's running skills and flanker Ashley Johnson's raw power to make headway against a well-organised Worcester defence, and a spell of pressure gained its reward when Goode slotted a 28th-minute penalty.
Clear try-scoring opportunities were at a premium, though, with defences cancelling each other out and ensuring an overall entertainment factor struggled to creep above zero.
Goode edged Wasps ahead with a second penalty shortly after the restart, and Worcester continued to be frustrated in their attempts to break down a visiting defence that had Johnson as its core figure.
But the try deadlock was finally broken after 53 minutes thanks to a piece of magic from Worcester's Samoa internatonal wing David Lemi.
Lemi's devastating footwork took him away from would-be Wasps tacklers, and Betty was on hand to finish things off wide out.
A Lamb penalty 18 minutes from time further eased Worcester's nerves, and Wasps could offer next to nothing in reply, lacking any attacking direction at half-back and being forced to rely on flashes of inspiration by Daly.
And it took an intervention from Wasps' tighthead prop, plus the reliable Goode's right boot, to grasp victory from the jaws of defeat.
- Sports & Recreation
- Andy Goode