The Irish province were already assured of a place in the last eight as one of the best runners-up following surprise defeats for Harlequins and Toulouse on Friday night.
A win would have seen Ulster top the pool and Morgan Parra and Ruan Pienaar both struck two penalties each as the sides went into the break tied at 6-6.
The pair traded penalties again at a noisy Stade Marcel Michelin before Pienaar put the visitors ahead again after 57 minutes.
But Ulster will be on the road in the quarter-finals for the second year in succession after Ti'i Paulo's try 15 minutes from time gave Clermont a decisive advantage.
Pienaar brought Ulster back to within a point but Parra's final penalty ensured Clermont finish as table toppers with a better head-to-head record in terms of tries scored over the two matches played.
Clermont are unbeaten at home since November 2009 and they knew that only another victory would prevent them from failing to make the knockout stages for a second consecutive year.
Ulster had beaten Clermont 16-11 at Ravenhill in November but the French side are a different proposition at home and they had Brian McLaughlin's men on the back foot inside the opening minute.
Wesley Fofana swept around two Ulstermen and found Aurelien Rougerie supporting on the right wing. With men over, a try looked certain but mercifully for the visitors the Clermont captain's difficult return pass was adjudged to be forward.
Fofana claimed to be impeded as he threatened down the right and former Leinster second row Nathan Hines was also close to getting through as Ulster struggled to hold on.
McLaughlin's charges finally got some respite when they swept to the opposite wing from a lineout but Andrew Trimble fumbled in a tight corner.
Momentum looked set to swing the home side's way again when Ulster were penalised and Parra took the penalty quickly.
He was caught high by Dan Tuohy, and when the Ulster lock was sin-binned, Parra picked himself up to put Clermont ahead after 14 minutes.
Ulster captain Johann Muller was leading by example in the lineout, though, and when he was dragged to the ground illegally, his South African compatriot Pienaar levelled things up.
Parra punished another infringement at scrum-time to push the hosts in front again after 22 minutes but Ulster almost got a break when a wild pass by Rougerie was cut out by Craig Gilroy. A desperate tap tackle by prop Lionel Faure was enough to down the young Ulster winger.
Pienaar missed a difficult penalty attempt but he was given another chance after 32 minutes when Trimble was held up in the right corner. This time his strike was true.
Julien Malzieu's dangerous break was then stopped by Ulster full-back Stefan Terblanche and Pienaar missed another long distance penalty. Terblanche came up short with an audacious drop goal attempt as the sides went in level at the interval.
A huge push by the Ulster pack saw them win a penalty on a Clermont scrum as Pienaar put Ulster ahead for the first time, six minutes into the second half.
The home side responded by introducing Vincent Debaty and Brock James and they were rewarded instantly as Parra punished another Ulster infringement.
Pienaar kicked Ulster ahead again on 57 minutes but they had a real let off when Lee Byrne's try was ruled out after Hines obstructed both Stephen Ferris and Pedrie Wannenburg.
It proved a short reprieve however as replacement hooker Paulo burrowed over following a sustained period of Clermont pressure on the Ulster line. However, referee Dave Pearson and his assistants missed a blatant block by Hines on Chris Henry which made it easier for Paulo to rumble over.
Parra converted to put Clermont 16-12 in front with 15 minutes left. Pienaar replied but Parra got the final score with four minutes remaining, stretching Clermont's brilliant unbeaten record at home to 37 matches.
Forwards Thomas Waldrom and Ed Slater were the star performers to ensure the Tigers had plentiful possession but without half-backs Toby Flood and Ben Youngs and centre Manu Tuilagi, the hosts had considerable difficulty in breaking down a stubborn Italian defence.
They led only 9-6 at the end of a desperately poor first half but with wind advantage in the second half the Tigers upped their game, scoring four tries to ensure Aironi finished pointless in the group, having conceded a massive 274 points in their six pool games.
The Tigers dominated the opening 10 minutes, choosing not to take two kickable penalties in favour of more attacking options, but careless handling prevented them from collecting any points.
Against the run of play, Aironi had the first chance for points and Naas Olivier made no mistake with his 50-metre penalty to give the Italians a 3-0 lead after 12 minutes.
Geordan Murphy, Tom Croft and Waldrom then constructed Leicester's best movement of the first half to send Alesana Tuilagi over the try line, but the score was ruled out as Croft was adjudged to have put a foot in touch.
Eventually, after 21 minutes, the Tigers got on the scoreboard when they elected to kick at goal and Billy Twelvetrees made no mistake with his 25-metre kick.
Olivier was given a chance to restore Aironi's advantage and again he was successful, this time with a wind assisted kick from 60 metres, but this was soon nullified by a second penalty from Twelvetrees.
Leicester continued to dominate the scrum, pushing Aironi back at every set piece, but still the Tigers couldn't pick up a first-half try and it was left to another penalty from Twelvetrees to give his side a 9-6 half-time lead.
Within three minutes of the restart, the Tigers finally crossed the line. From a ruck 20 metres out, a pass from Mickey Young sent Waldrom through a huge gap in the visitors' defence for the number eight to score unopposed.
Twelvetrees missed the conversion, sending his simple kick wide.
Four minutes later and Leicester were over again as the Aironi defence was now being stretched to the limit. Two strong driving runs from Slater put the Italians on the back foot and Ben Woods was on hand for the try which Twelvetrees converted.
After 57 minutes Leicester scored their third try when Murphy dummied his way over to give his side a 26-6 lead going into the final quarter, but still poor handling in the difficult conditions continued to blight the Tigers' performance.
It took Leicester until the 71st minute to collect their bonus point when replacement George Ford displayed some neat footwork to fox the Aironi defence for a try which he also converted.