Billy Twelvetrees scored all of the Tigers' points with a pair of tries, two conversions and five penalties to at least secure the visitors a losing bonus point.
Having beaten reigning champions Saracens at Twickenham last weekend, Wasps made it a dream start for new rugby director Dai Young by beating the 2010 champions.
Wasps fielded five home debutants in Hugo Southwell, Chris Bell, Nicky Robinson, Marco Wentzel and Jonathan Poff. But scrum-half Charlie Davies failed a fitness-test an hour before kick-off and was replaced by Nic Berry.
Leicester, beaten at home by Exeter last weekend,
continue to function under the handicap of being without 11 players who are on World Cup duty, and five out injured including Jordan Crane, Craig Newby and Julian White.
But they put Wasps under pressure from the kick-off, and when flanker Poff went off his feet at the breakdown, Twelvetrees opened the scoring with a penalty.
Former Wales fly-half Robinson had the opportunity to equalise on nine minutes, but shanked his shot.
Robinson made amends on 12 minutes, his penalty soaring high and turning in late to equalise.
Twelvetrees struck again on on 15 minutes after Wasps were penalised at a scrum. The centre's penalty virtually corkscrewed in flight so it swung out wide then turned in late.
Twelvetrees then turned striker, finishing a neat passing sequence by scoring on the right from a pass outside by Niall Morris. He forced his way past Robinson and Varndell to twist over the line and completed the job by converting from wide out.
Robinson was unable to match such expertise on 22 minutes, his long-range penalty sailing wide to leave Wasps 10 points adrift.
Varndell looked dangerous whenever he was on the ball, and he hunted down full-back Scott Hamilton after kicking ahead and forced him to concede a penalty. Wasps opted for a lineout which Tigers cleared.
Wasps were rewarded for enterprising rugby on 27 minutes. Southwell broke out from deep, Wade's pace on the right left two would-be tacklers trailing, and his pass inside sent Berry scooting clear for a try which Robinson converted.
Wade then scored a fine try of his own on 35 minutes. Varndell did the damage up the left, sucking in three tacklers to haul him down. Wasps sped the ball right via three passes, the last of which sent Wade racing home, Robinson again converting.
Back stormed Tigers, fly-half George Ford making an incisive break off the back of a ruck and passing inside for Twelvetrees to score by the post and convert his own try.
Robinson thought he had levelled on the stroke of half-time, but his penalty from wide right struck the near post and bounced wide.
Twelvetrees continued his points-fest five minutes after the break with a penalty, but Wasps replied with one of the tries of the season so far.
There seemed no more than moderate danger when Southwell passed right to Wade deep inside their own half. But the young wing twice shaped to pass only to accelerate into the Tigers half and swerve outside two challengers for a memorable score.
Robinson's conversion hit the post - his fourth failure from seven shots at goal - and he limped out of the game seconds later, replaced by Ryan Davis who fired his team in front on 53 minutes with his first penalty.
The ruthless Twelvetrees regained the lead for Leicester with a 64th-minute penalty after Wasps went offside in front of their posts.
But Davis kept the see-saw effect alive, scoring on 69 minutes, the fifth time the lead changed hands.
And when Wade darted behinds two defenders to win a race for a box-kick from Berry, victory was secured.
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