Thomas Waldrom scored the first points of the game when he crossed over to put the visitors in front with less than 10 minutes gone.
Fly-half Toby Flood put another couple of points on top of the try before adding a penalty to his tally in a matter of minutes.
Opposite number Nick Evans struck back, scoring four first-half penalties of his own and converting two Quins tries to fire his side in front.
Flood responded with his second and third penalties of the afternoon, while convering substitute Steven Mafi's five points to close the gap to three points at the half.
The Leicester fly-half added yet another penalty to his tally almost immediately after the restart, tieing the score at 26 points apiece.
However, Quins center George Lowe struck back in minutes to score the first five points of the second half to put his side back in front. Evans' ensuing conversion extended their lead to seven.
Leicester's Alesana Tuilagi crossed over with 15 minutes to go, allowing Flood to level the score, before Waldrom gave the visitors a late lead with another five points. Flood's conversion and final three points of the game then sealed the comeback win.