Ulster beat Montpellier 27-16 on Friday but failed to grab a crucial fourth try which would have guaranteed them a European Cup quarter-final place.
Montpellier's Figian player Timoci Nagusa (L) hangs on to Ulster's Irish player Roger Wilson (R) during the European Cup rugby union match between Ulster and Montpellier at Ravenhill Stadium in Belfast, Northern Ireland on January 10, 2014
The Irish province racked up a fifth win in five Pool 5 matches to give them a seven-point lead over Leicester.
However, the English giants should close the gap with a win at group whipping boys Treviso on Saturday before a potential pool decider at Welford Road against Ulster next weekend.
Ulster had looked in control, but never particularly comfortable in the first 40 minutes at Ravenhill against a Montpellier side which had lost six of their last seven matches.
Having slipped down 3-0 to a seventh-minute penalty from Eric Escande, Ulster were swiftly on top with two tries in three minutes.
Paddy Jackson's accurate cross-field kick was calmly collected by lock forward Robbie Diack to dive over for his third career try in the tournament.
South African scrum-half Ruan Pienaar kicked a difficult conversion from tight on the touchline for a 7-3 lead.
Pienaar added his team's second try on 26 minutes after collecting his own kick ahead.
His simple conversion made it 14-3 but Ulster took their foot off the gas and ended up going in at just 17-13 in front at the interval.
Montpellier lock Robins Tchale-Watchou burst through for 14-8 with Escande adding the extras.
Pienaar and Escande then exchanged penalties before half-time.
A penalty apiece followed early in the second half before former All Blacks prop John Afoa barrelled through for Ulster's third try just before hour mark.
Pienaar added the conversion for a personal haul of 17 points on the night but some dogged defence from Montpellier prevented Ulster from grabbing the bonus point try.
In the night's other match, Scarlets beat Racing Metro 19-13 in Pool 4, but neither side can make the last eight.
"It wasn't a great match," said Scarlets fly-half Rhys Priestland who kicked 14 points.
"We put in a lot of hard work this season. We didn't have the precision required but we showed a lot of determination."