The champions looked set for a fourth successive defeat after a dire first half.
But fly-half Staunton’s boot steadied the ship and a try from Damien Varley sent the Tigers crashing to their first defeat of the campaign.
Wasps’ man-of-the-match Tom Rees said: “That was badly needed to get our season back on track.
“After a disappointing first half we pulled ourselves together and showed the real Wasps’ character — and it’s essential now that we kick on and build on this.”
Leicester boss Heyneke Meyer said: “That was an extremely disappointing second half. I don’t like excuses but eight or nine top-class players are not available at the moment.
“But we have a good squad with some good young players coming through and had we made the most of our first half superiority it would have made for a different result.”
Wasps’ first-half performance had certainly been one of their worst in recent years.
They struggled in the scrum, missed tackles and looked decidedly wobbly under the aerial bombardment that Leicester launched during a period when they should have sewn up the points.
No sooner had Wasps switched Josh Lewsey to partner Paul Sackey on the wing for recent fumbles than new full-back Tom Voyce developed Teflon hands.
Incredibly, the scores were locked at 6-6 after half an hour with two kicks apiece from Tigers’ Toby Flood and Staunton.
Leicester’s superiority finally told on 34 minutes.
Aaron Mauger smashed through a tackle by Staunton and sent winger Johne Murphy in for a try by the posts, Flood converting.
But a half-time roasting from head coach Shaun Edwards transformed the champions and Ireland fly-half Staunton set the alarm bells ringing at Welford Road.
A penalty two minutes into the second half was matched by one from Flood.
But then Staunton had the vision to arrow home two drop goals in three minutes that turned the game.
Leicester’s confidence drained away in front of the 17,000 capacity crowd and old hatreds between these two erupted on 54 minutes.
It was sparked by Julian White, who sprawled all over former All Black Mark Robinson.
To further enrage the crowd, referee Wayne Barnes ordered uncontested scrums once the Wasps props Phil Vickery and Pat Barnard both left the field injured.
Given the number of times Wasps have done this, it was no surprise that the crowds roared “cheats, cheats”, but the laws left the match official with no choice.
Flood nudged Leicester back in front on 63 minutes with a fifth kick but Staunton rifled over another penalty and the match was sealed on 68 minutes when Varley squeezed the ball down under a mountain of bodies and ref awarded it after consulting the video ref.
It completed a remarkable turnaround. The last time these teams met Wasps beat Leicester 21-16 in the Guinness Premiership Final but after their first-half show you would not have backed them to beat a rug.