Quins were marginally the better side for much of the contest but aside from an Ugo Monye try that put them in front early on, they could not break down a stubborn visiting defence.
The hosts led with 12 minutes remaining but the Chiefs, inspired by fly-half Gareth Steenson, produced two late tries from Damian Welch and Luke Arscott as their bid to qualify for next season's Heineken Cup was given a boost.
The result means Exeter have done the double over the Quins after they also defeated the champions 42-28 at Sandy Park in October, while Harlequins' slip means Saracens can replace them at the summit with victory over London Welsh on Sunday.
Leicester got their bid for a home play-off semi-final back on track by thrashing Sale 48-10 at Welford Road.
The five points from the six tries to one victory was a major boost for Leicester after successive defeats by the top two clubs, Harlequins and Saracens.
Leicester dominated the bottom club from start to finish, and there were two tries each for lock Ed Slater and centre Matt Smith, one from full-back Mathew Tait and a penalty try.
Fly-half George Ford finished the match with a 100 per cent kicking record, converting all the tries and kicking two penalties for a tally of 18 points.
Replacement Andy Powell scored Sale's only try with Danny Cipriani kicking a conversion and Nick Macleod a penalty.
Jamie Elliott ran in a hat-trick as Northampton moved into the play-off places with a six-try 40-14 demolition of London Irish at Franklin's Gardens.
The Irish backline was a constant threat in the first period, but the power of the Northampton pack took over after the break as the home side scored three tries in 15 minutes either side of half-time to settle the contest.
Elliott's hat-trick makes it five tries in two games for the winger, while Samu Manoa, Soane Tonga'uiha and Gerrit-Jan van Velze also crossed for Northampton.
Guy Armitage scored a consolation try for the Exiles, but they will have rued some missed opportunities before the home pack built up a head of steam.