Quins do damage before half-time
Harlequins protected fourth place in the Aviva Premiership but were unable to produce the bonus point that looked certain following a dominant first half.
A crowd of 74,827 at Twickenham saw Nick Evans, Charlie Walker and Mike Brown score tries as the 2012 champions entered the interval 19-3 ahead.
But they could only then manage a penalty from Evans as the entertainment produced in the first half of 'Big Game Six' was replaced by an arm wrestle that finished 3-3 on the scoreboard.
Former All Blacks fly-half Evans endured a mixed afternoon, kicking two conversions and a penalty but missing eight points from the kicking tee.
Hooker Dave Ward finished man of the match after proving a constant thorn in Exeter's side at the breakdown and there were also notable performances from centre Matt Hopper and wing Charlie Walker.
Exeter's hopes of replacing them in fourth may have been foiled, but director of rugby Conor O'Shea will be ruing the failure to claim all five points.
The Chiefs became more successful at disrupting their fast-paced offloading game, but a dominant third quarter deserved better than a solitary Evans penalty.
A breathless opening signposted the action to come in a high-energy first half that saw both sides attack at every opportunity.
It took just over two minutes for Quins to run in the first try after a turnover from Ward launched the attack just as Exeter had offered an overlap on the left.
Quins exploited it in a flash with quick hands from Nick Easter and Luke Wallace sending Evans racing over.
Exeter fly-half Gareth Steenson landed a penalty after Matt Hopper had tip-tackled Jason Shoemark and the Chiefs were playing with an ambition that was unrewarded on the scoreboard.
A break from their own 22 ended with Luke Arscott kicking the ball away aimlessly, Dave Ewers surged through the line only to concede a penalty and Shoemark failed to pass having sped clear.
In contrast Quins were more clinical and after Tom Casson and Brown had been kept a bay with desperate defending, they added two fine tries.
Exeter's line was stretched as Easter spearheaded an attack down the left wing and Walker finished in the opposite corner after stepping Tom Johnson.
The try came with the added bonus of Jack Nowell being sin-binned for a tip-tackle on Hopper, although Exeter will feel aggrieved that Hopper escaped a similar punishment for his near-identical offence earlier on.
Just seconds of the first half remained when Quins struck for a third time, Care darting through a gap at the breakdown following another steal from Ward and sending Brown over.
A long-range penalty from Steenson shortly after the interval offered Exeter a glimmer of hope, but they lacked the firepower to mount a serious comeback.
The second half was more attritional than the first and it was Quins who were winning the slugfest as they camped deep in opposition territory in search of the bonus point.
Arscott made a key tackle to stop a try in the left corner and Quins finished with three points from Evans to restore an unassailable lead.
Exeter controlled the final 15 minutes as they searched for a bonus point of their own and would have crossed had Arscott's pass been more forgiving.
A succession of penalties and set pieces followed until referee JP Doyle finally blew the whistle and the dying moments saw Quins search in vain for the fourth try.