Lee Thomas came off the bench at half-time at Adams Park to kick two penalties which secured Wasps a narrow victory.
"We're really pleased to get the win," said director of rugby Young after Wasps denied the Tigers the chance to go top of the Aviva Premiership.
"In reality, we should have won that game more comfortably than we did.
"Leicester, being Leicester, just don't go away but I don't think they threatened us that much.We always felt pretty comfortable in defence."
Former Leicester winger Tom Varndell set Wasps on the way to victory with a third-minute try - the eighth this season for the Premiership's leading tryscorer - and 21-year-old winger Christian Wade was stopped a metre short after a dazzling run from the halfway line.
But Young, who praised centre Elliot Daly as "a threat every time he had the ball," was equally impressed by the performance of his pack lacking England locks Joe Launchbury and Tom Palmer.
"I thought the two young props - Simon McIntyre and Phil Swainston - did really well and James Cannon in the second row stood up."
Young, relieved that fly-half George Ford missed a late penalty which would have won the game for Leicester, added: "I would have taken that before the game.
"But to be sitting here picking holes because I'm disappointed that we didn't make more of the opportunities is a positive thing for us.
"We have to be a little bit more accurate, ruthless and clinical to win these games because I don't want to go through that every week."
Tigers director of rugby Richard Cockerill said: "We're disappointed we didn't win. It was a pretty poor game from both sides - a pretty dogged, horrible non-event.
"With international call-ups and a couple of injuries we had about 16 or 17 players but we should have played better.
"We got no set-piece parity at times. In the second half the scrum was diabolical, whether it was us, them or the ref. We made it hard on ourselves.
"We didn't deliver from the scrum or the line-out. Our accuracy and delivery were really poor around the set-piece."
Cockerill admitted that Wasps showed more attacking flair than his side.
"They have some very quick guys - Elliot Daly and the two wingers - and they put us under some stress in the wide channels," he added.
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