The win puts Leicester, who started the day second bottom of Pool Two, on nine points and still in the hunt with just one match left.
They go away to Scarlets next week, aiming to top the group and reach the semi-finals.
Leicester's tries came through Logovi'i Mulipola, Michael Noone, Mathew Tait and Vereniki Goneva, who crossed twice, with Dan Bowden kicking a conversion and a penalty and Matt Cornwell two conversions.
Wasps, who have not won at Leicester since 2008, were outplayed for most of the game and scored a consolation try through replacement Hugo Southwell near the end to add to a penalty from Tommy Bell.
Leicester, even without 18 players on international duty, were able to field a side boasting internationals such as Tom Croft, Scott Hamilton and Goneva, while Wasps were virtually unrecognisable with five players making their debuts.
England flanker Croft was making only his second start since returning from a shoulder operation, which kept him out for more than eight months, but he lasted only 11 minutes before going off injured and being replaced by Noone.
Wasps dominated possession early on, so much so that it took Leicester the best part of 10 minutes to get their hands on the ball properly, although fly-half Bowden did make a horrible mess of a fifth-minute penalty from 40 metres.
Leicester took the lead in the 18th minute when Samoa prop Mulipola drove over from a line-out, with Bowden adding the touchline conversion.
Leicester suffered another injury blow on the half hour when 19-year-old lock Tom Price, who was making his debut, was carried off after receiving almost 10 minutes of treatment for what appeared to be a neck injury suffered at a ruck. He was replaced by Harry Wells.
When play resumed, Bowden kicked a penalty to put Leicester 10-0 ahead.
Wasps flanker Tinus Du Plessis was sin-binned five minutes from half-time for a professional foul at a ruck and Leicester took full advantage of the extra man by scoring from a line-out.
They drove to the line and Bowden caught Wasps napping with a great diagonal run down the blind side to create an easy try for Noone. Bowden missed the conversion.
Bell sparked a period of pressure for Wasps at the start of the second half but they spurned a decent chance by ignoring an overlap.
Bell, however, put Wasps on the scoreboard with a 47th-minute penalty, and the visitors then repelled a promising Leicester attack by destroying their scrum.
Leicester centre Andy Forsyth created a great chance for winger Hamilton but the Kiwi's inside pass near the line was intercepted.
With the game starting to open up, Leicester's former England back Tait started to have a bigger influence and he finished off a fine Leicester move with a 20-metre dash for the line. Bowden missed the conversion.
Goneva scored Leicester's fourth try, converted by Cornwell before, with five minutes remaining, replacement Southwell salvaged some pride for Wasps with a try. Bell missed the conversion.
Goneva scored Leicester's fifth try, again converted by Cornwell.
- Sports & Recreation
- Dan Bowden