Warriors fly-half Joe Carlisle scored the only points of the game - a penalty in each half - as his side put pressure on the home side throughout a wet and error-strewn match.
With their heaviest defeat of the season, a six-try 39-10 hammering by Gloucester on Boxing Day, still fresh in the memories, Wasps director of rugby Dai Young stated clearly in his programme notes: "We need a win."
His team dominated possession and were arguably more creative but continually mishandled the wet ball, failing to score in a match for the first time since they faced Sale in 2008.
With heavy rain making footing difficult, four players suffered injuries before the end of a first half that lasted a marathon 51 minutes.
Carlisle - who had earlier hooked wide right with a long-range attempt - finally opened the scoring with the final kick of the half, converting a penalty from the left touchline.
Wasps went into the match with nine players on their injury list and stand-in full-back Richard Houghton did not even make the opening kick-off, hurting his groin during the warm-up.
Samoa number eight Ezra Taylor's debut for Worcester lasted just two minutes and five seconds before he had to be stretchered from the field, in obvious pain with a serious knee injury which was later revealed to be a ruptured patella.
Despite spending much of the first quarter of the game pressuring the Worcester 22-metre line, Wasps could not make that territorial advantage tell.
The best chance came when right wing Christian Wade ran onto a cross-field kick from Nick Robinson but spilled under pressure from opposite number Miles Benjamin less than five metres from the try line.
Neither Wade nor Robinson made it to half-time as both succumbed to injury, the latter nursing a troublesome left calf.
Wade was helped from the field, limping on his left ankle after going to ground with two Warriors on top of him. That killed the momentum of a passage of play set up by a bullocking run up the middle from Namibian back-rower Tinus du Plessis, making his debut for the home side.
It brought former Northampton Saint Chris Mayor onto the field, only half an hour after he was drafted onto the bench due to Houghton's last-minute withdrawal.
Worcester were down to 14 men early in the second half when referee Wayne Barnes - in charge for a 100th Premiership match - brandished the yellow card for Marcel Garvey's trip on Tom Varndell.
Second-choice goal-kicker Ryan Davies' penalty dropped short, though, and the decision to use left-footer Elliot when a penalty was given on the right in the 64th minute backfired as he hooked well wide of the posts. Davies had already failed with a drop-goal attempt early in the half.
Worcester moved ahead of the home side in the league table, while Wasps dropped to second from bottom and must now regroup in time for a tough trip to Leicester next Saturday.
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