Two tries in a seven-minute spell early in the second half kept Leinster's hopes of remaining in the Heineken Cup alive, but they were forced to battle hard against newcomers Exeter.
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The reigning champions will have to wait until Sunday to see if they or Munster, who welcome Racing Metro, progress to the quarter-finals as one of the best second placed sides alongside Montpellier.
Exeter went in to the break with a 17-12 lead after a penalty try plus a touchdown from Neil Clark with Gareth Steenson converting both and adding a late penalty. Leinster replied with tries from Gordon D'Arcy - converted by Johnny Sexton - and Rob Kearney.
Leinster regained the lead early in the second half with a try from Brian O'Driscoll, converted by Sexton - who then added the extras to the bonus point try which followed quickly through Ireland skipper Jamie Heaslip.
Steenson replied with a penalty which was cancelled out by Sexton before the Chiefs had Ben White sent to the sin bin in the closing minutes, while Leinster's Mike Ross was also yellow carded soon after.
Exeter fell behind after just three minutes of the match. They conceded two penalties, which allowed the visitors to pressure the line and resulted in centre D'Arcy crossing for the opening try.
Assistant referee Eric Soulan had to be replaced after a collision, but more worrying was the sight of Exeter's England international flanker Tom Johnson being carried off on a stretcher. With the RBS 6 Nations approaching, it could be a blow for club and country.
The hosts were back on level terms when, with nothing to lose, they kicked a penalty to touch. Australia international Dean Mumm set up the drive from the lineout which led to hooker Clark touching down in the corner and fly-half Steenson adding the extras.
Just before the half-hour mark Leinster's front row appeared to stand up but the referee allowed play to continue. The visitors worked the ball wide to maximise their numbers, allowing fullback Kearney to crash over.
Exeter turned down a simple penalty from in front of the posts and, when Leinster pulled down the 15-man maul after the lineout, French referee Romain Poite awarded the Chiefs a penalty try.
Steenson converted and then just before half-time he added a 38-metre penalty to give the Chiefs a surprise 17-12 interval lead.
Leinster changed their tactics at the start of the second half, testing Exeter with the high up-and-under as they increased the intensity. They regained the lead as O'Driscoll went over for their third try, converted by Sexton.
Minutes later they went in search of the bonus-point try, kicking a penalty to the corner to set up a drive to the line. Exeter were again penalised and although they won the turnover ball at the next drive they lost the ball at their scrum.
Leinster maintained the pressure but the Chiefs knocked on close to their own line. From the scrum number eight Heaslip picked up at the base and saw a gap open up in front of him as he crossed the line, with Sexton converting for a 26-17 lead.
Steenson added his second penalty to reduce the arrears to six points and set up a potentially exciting finale.
But then the Chiefs were forced to play the last seven minutes with 14 men when replacement White was yellow-carded.
Sexton added a penalty but then Leinster were also reduced to 14 men when prop Ross was sent to the sin bin. Exeter maintained the pressure but the clock was against them as they bowed out of the competition.