Harlequins had only beaten Leicester once before in 26 Premiership encounters - at The Stoop back in April 2003
Undefeated Premiership leaders Harlequins inflicted a fifth loss on Leicester and recorded a first league win at Welford Road since 1997.
The game got off to a blistering start as Boris Stankovich scored for Tigers and Quins replied through Sam Smith and Seb Stegmann, all inside eight minutes.
Scott Hamilton went over for Tigers before both sides exchanged penalties to give Quins a narrow half-time lead.
More penalties were exchanged after the break before Ollie Kohn's decisive try.
The hosts took an early lead in classic Welford Road style as a catch-and-drive put Stankovich over.
Quins' response was swift as Smith finished a simple blind-side move and a minute later the visitors worked the ball around the Tigers defence to put Stegmann clear.
Evans converted both scores to make it 14-5 to Quins.
Leicester's backs held some wayward passes to put Hamilton over in the corner as Staunton converted to bring Tigers back to just 14-12 down.
Alesana Tuilagi thought he had scored in his first game back from the World Cup but replays showed Jordan Turner-Hall's tackle had put the Samoan in touch.
Tigers started to gain dominance but Quins hung on and after Staunton converted a penalty to give Leicester a one-point lead Evans struck back immediately to make it 17-15 to his side at half-time.
Sam Smith fumbled the first good chance after the interval, following some great offloading by Quins, and both sides struggled to gain the upper hand in the face of stiffer defending across the pitch.
Evans extended Quins' lead through a penalty with half an hour left but no sooner had the New Zealander done so that Staunton pulled one back for Tigers when the Londoners were offside at a ruck.
Evans, and then Staunton, missed reasonably simple penalty attempts but with five minutes remaining Quins mauled their way over from 30m out and Evans landed the conversion.
Leicester director of rugby Richard Cockerill:
"We've got to defend better. It was those two phases that killed us.
"We thought we could win today but our ball-carrying and decision-making was poor and we lost lineouts at crucial times.
"It's been a farce with 20 guys missing, it's been very hard but we will be stronger for it.
"Whilst I'm sorry England lost [to France in today's quarter-final], the result in New Zealand is a delight for the club. Europe is big for us and we need the players to be back and integrated by then."
Harlequins director of rugby Conor O'Shea:
"We won't get carried away but I'd rather be in our position than Leicester's. It was an incredible ask for them [as so many of their players are at the World Cup].
"We were under the pump in the first half but we were very opportunistic and then in the second half we dug in and I am really proud of the guys.
"Last season we lost 10 league games by one score or less, this season we are playing with more composure and the confidence that comes from winning."
This page is best viewed in an up-to-date web browser with style sheets (CSS) enabled. While you will be able to view the content of this page in your current browser, you will not be able to get the full visual experience. Please consider upgrading your browser software or enabling style sheets (CSS) if you are able to do so.