Olly Woodburn scored two tries as Exeter pulled off a stunning 35-26 victory in a pulsating contest as Leicester missed the chance to climb to second in the Gallagher Prem table.
Leicester lost ground in their attempt to secure a home semi-final as Henry Slade's last-minute penalty prevented them from taking anything from the match, but they are still safely into the play-offs.
With Bristol Bears' hopes of making the top four now over, fourth-placed Exeter's match against fifth-placed Saracens at Sandy Park on Saturday will decide the final team in the play-offs. Exeter are three points ahead of Saracens in the table.
Exeter were deserved winners by scoring four tries to three, with Len Ikitau and Andrea Zambonin also touching down while Slade kicked three penalties and three conversions.
Exeter forwards coach Ross McMillan said: "We were frustrated by our performance at Harlequins last week, but throughout the season every time we've had a setback we've bounced back.
"The tight squad have great character and have always responded when they've been knocked around a bit.
"We knew it would be a tough challenge from the outset but the boys took it on board and I loved the battles that went on today, especially the scraps on the floor and in the air as possession of that ball is vital.
"The Leicester pack are used to dominating sides here so I'm incredibly proud at how we fronted up to them."
Leicester's tries came from George Pearson, Olly Cracknell and Charlie Clare while Orlando Bailey added 11 points with his boot.
Hanro Liebenberg led out Tigers for his 150th appearance for the club and his side were soon ahead when Bailey kicked a fourth-minute penalty.
But almost immediately from the restart, Immanuel Feyi-Waboso collected a clearance to sail past three defenders before timing his pass perfectly to send Woodburn in under the posts.
Slade converted and added a straightforward penalty before Tigers replied with an impressive score when Pearson collected a superbly judged cross-field kick from Bailey.
Leicester looked to have regained the lead when a break from Bailey saw Adam Radwan cross the line but TMO replays showed the wing had been held up by the combined efforts of Slade and Harvey Skinner.
However Tigers were not to be denied and a second penalty from Bailey put them back in front but this was soon nullified by one from Slade.
Leicester blew a golden opportunity when Ollie Hassell-Collins failed to collect a poor pass from Pearson with the line at his mercy after strong runs from Jack van Poortvliet and Jamie Blamire had put the defence on the back foot.
But the hosts went into the interval 14-13 ahead thanks to another Bailey penalty.
Three minutes after the restart, Exeter scored an excellent second try when a neat off-load from Dafydd Jenkins sent Tom Hooper racing into the opposition 22 and when the ball was recycled, Ikitau out-flanked the cover to score.
Leicester changed four forwards in one swoop but it did not pay dividends as Exeter immediately extended their lead when Woodburn took an inside pass from Ikitau for his second try.
However Woodburn was then sin-binned for a deliberate knock-on with Tigers capitalising when Cracknell powered over to leave his side trailing by four points going into the final quarter.
Woodburn returned from the sin-bin with no further damage done to the scoreboard but on his return Clare finished from close-range to set up a tense finish.
A handling error from the home side enabled Exeter to regain the lead when a long pass saw Zambonin evade a weak challenge from Adam Radwan for the bonus-point try with Slade's last-minute penalty leaving Tigers with no points from the game.
