The home side started poorly, going 6-0 down in the opening minutes as George Ford missed two penalties for Tigers.
Ford got Leicester off the mark after 17 minutes, and levelled the match with a penalty just before half-time.
Matt Smith scored the only try of the tie at the start of the second half to put Leicester ahead for the first time.
Two more missed penalties could have cost the Tigers late on as Saints reduced the gap to four points.
However, a yellow card for Ryan Lamb in the 77th minute left the visitors a man short for the rest of the game.