Leicester wing Chris Ashton became the first player to score 100 Premiership tries as Exeter conceded their record points tally in a top-flight match.
Veteran Ashton, who retires at the end of the season, scored a hat-trick to help Tigers demolish 14-man Chiefs.
The 36-year-old went close to a fourth score - but TV replays prompted referee Karl Dickson to award a penalty try.
Olly Woodburn was shown a second yellow card at the same time, virtually ending Exeter's faint play-off hopes.
With two games left, they are now 10 points adrift of fourth-placed Northampton, while Leicester all but sealed a top-four spot and still have an outside chance of a home semi-final in the play-offs.
Despite losing tight-head prop James Cronin to injury early in the game, the Tigers built up a head of steam as they raced into a 14-point lead with two converted tries inside the opening 10 minutes.
Dan Kelly pounced on Handre Pollard's grubber for the first and Jack van Poortvliet - who had just come close to scoring - burst off a scrum and found Ashton, who dabbed a kick in behind for his team-mate to dive over.
Exeter responded well, Sam Simmonds exploiting a gap in Tigers' blind-side defence to finish superbly after a weaving run to the line.
Pollard's kicking was variable, with a routine penalty crashing back off the post after Woodburn had received his first booking for a deliberate knock-on.
When the fly-half's careless cross-kick was neatly plucked out of the air by Sean O'Brien, Stuart Hogg took advantage to score a try that reduced the deficit to five points - but the short-handed visitors were unable to build on that momentum.
As Woodburn returned to the field, Rory O'Loughlin replaced him in the sin-bin for an identical offence - and Anthony Watson's grubber kick released Ashton in the corner for his first to make it 22-12 on the stroke of half-time.
Ashton might have added a second soon after the resumption as he slid in to finish off Jasper Wiese's break, but appeared to have a foot in touch under pressure from Hogg's fine last-ditch tackle.
After lengthy consultation with the TMO though, referee Karl Dickson awarded Leicester a penalty try, deciding Woodburn had dived onto the back of Ashton, earning the Exeter winger a harsh second yellow card, which reduced the visitors to 14.
Five minutes later, Ashton did bring up his century, collecting Pollard's pass to go over on the opposite flank.
With a sixth straight league win all but confirmed, the Tigers then ripped their demoralised opponents apart.
George Martin burrowed over for Leicester's sixth try and Matt Scott's double sandwiched a sixth Premiership hat-trick try for Ashton, who finished off a flowing move to take his tally to 101.
The celebratory atmosphere at Welford Road continued as Ben Youngs came off the bench for his 300th Leicester appearance and even a late consolation for the Chiefs, a close-range effort by Rus Tuima, barely registered with the jubilant home crowd.
Leicester: Steward, Ashton, Scott, Kelly, Watson, Pollard, Van Poortvliet, Cronin, Montoya, Cole, Martin, Henderson, Liebenberg, Reffell, Wiese.
Replacements: Clare, West, Heyes, Snyman, Cracknell, B Youngs, Gopperth, Potter.
Exeter: Hogg, Wyatt, O'Brien, Kata, Woodburn, S Simmonds, Becconsall, Abuladze, Yeandle, Iosefa-Scott, Dunne, J Gray, Davis, Tshiunza, J Simmonds.
Replacements: Innard, Sio, Street, Williams, Tuima, Townsend, Skinner, O'Loughlin.
Referee: Karl Dickson