Northampton Saints' victory at Newcastle Falcons saw it leapfrog Harlequins and the now fifth-placed side could only respond by collecting a losing bonus point in a 17-15 defeat.
Exeter made a flying start and seized the outright record for consecutive matches with tries as Ollie Devoto dotted down inside two minutes of game 72.
Nic White went through soon afterwards and the Chiefs' lead was extended further by Harry Williams's try early in the second half.
Harlequins belatedly responded but it was only enough for a point as Sam Hidalgo-Clyne and Charlie Walker crossed in quick succession.
Second-placed Saracens won 31-14 at Wasps thanks to first-half tries from Ben Spencer, Liam Williams and Will Skelton.
Owen Farrell also kicked four penalties in a one-sided affair.
Meanwhile, Leicester Tigers is not yet entirely safe from relegation because of a painful late 23-20 home reverse at the hands of Bristol Bears.
Jordan Olowofela scored Leicester's only try after half an hour, but George Ford's kicking kept them in front until Harry Thacker crossed for Bristol's third score five minutes from time.