The Welford Road hosts rescued their campaign from the perils of possible pool stage elimination.
The Tigers, European champions in 2001 and 2002, appeared to have little chance of progressing from Pool Four when Clermont cruised into a nine-point interval lead.
But Leicester rapidly wiped out that deficit before Toby Flood's 62nd and 72nd-minute penalties secured the points and kept Tigers on course for an 11th quarter-final appearance in 15 seasons of Heineken Cup rugby.
England centre Manu Tuilagi and flanker Julian Salvi scored tries for the home side, while Flood kicked 13 points to put Leicester above Clermont in the group and just behind leaders Ulster.
Clermont's ex-New Zealand wing Sitiveni Sivivatu touched down for the visitors, while Morgan Parra and David Skrela both dropped goals, with Parra also slotting two penalties and a conversion.
But it was not enough during a game when the lead changed hands five times as Leicester narrowly staved off suffering a first European home defeat in five years since Munster beat them 21-19.
Leicester, Ulster or Clermont could progress as group winners when the tournament resumes next month, with Tigers travelling to Belfast before concluding the group at home against Italian strugglers Aironi.
Leicester boss Richard Cockerill made two changes from the side crushed 30-12 by Clermont at Stade Marcel Michelin last weekend, which represented Tigers' heaviest Heineken Cup defeat for eight years.
Prop Boris Stankovich and number eight Thomas Waldrom both received call-ups to the pack, while Clermont showed seven switches, including starts for ex-Scarlets centre Regan King and former Scotland captain Jason White.
Clermont had won only seven of their previous 28 Heineken Cup away games, but they went ahead inside two minutes when Skrela dropped a goal after an error-strewn start by Leicester.
Scrum-half Ben Youngs' pass to Flood went over his half-back partner's head, and the Tigers could not clear their defensive lines, Clermont maintained pressure and Skrela struck from close range.
Flood then missed his opening penalty chance, which did little to calm Leicester's nerves, before Parra emulated Skrela with a drop-goal strike that put Clermont six points clear.
Leicester needed a rapid response, and they produced one within 60 seconds through a simply-executed try as Flood's pass found Tuilagi, whose power and angle of attack easily took him through the Clermont midfield defence.
Flood converted for a 7-6 advantage, but Parra's opening penalty edged Clermont back in front as one former Scotland international replaced another when Nathan Hines took over from an injured White.
Skrela followed White in making an early exit through injury 10 minutes before the break - Brock James took over at fly-half - and Clermont continued to dominate possession and territory, forcing Leicester into making mistakes under pressure.
James and Flood both missed penalties as the interval approached, yet Clermont - with their Wales international full-back Lee Byrne kicking prodigiously for position - maintained momentum.
And they struck courtesy of a sweeping move that was sparked from turnover ball, with Parra and King involved before Byrne delivered a scoring pass to an unmarked Sivivatu.
Parra converted from wide out, and Leicester had a mountain to climb, trailing 16-7 midway through the contest.
Their response, though, was immense as they scored 10 points in four minutes to overhaul Clermont with more than half-an-hour of an enthralling contest remaining.
Flood booted a penalty before Youngs broke clear in open play and sent the supporting Australian openside Salvi over for a try that Flood improved.
It was a stunning riposte by the Tigers, but it soon became a case of backs to the wall again when Parra slotted another penalty that restored Clermont's lead approaching the end of an intense third quarter.
But the final 20 minutes belonged to Leicester, and two further Flood penalties sent Clermont home a beaten team, although their losing bonus point could still prove important when the group reaches its conclusion.
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