In a thrilling tie, two-time European champions Leicester edged the try count five to four in France and collected an offensive bonus point.
Matt Scott, Harry Simmons and Dan Kelly all scored first-half tries for the Tigers.
Scott and then James Cronin, in the last minute, added further scores after the break.
"We are obviously proud of our victory tonight against a massive team like Clermont and in an absolutely incredible atmosphere," said Leicester scrum-half and skipper Ben Youngs.
"It's a big performance to score 44 points here, but in the second half we suffered a lot more."
South African international stand-off Handre Pollard kicked 14 points on his first start for Leicester in the tournament.
Bautista Delguy, Etienne Fourcade, Anthony Belleau and Paul Jedrasiak grabbed tries for Clermont.
The French side were undone by indiscipline, conceding three times more penalties than Leicester.
"It's really very difficult to accept conceding 44 points at home, there is a bit of shame in that," said Clermont coach Jono Gibbes.
"Their ability to score points too easily against us was very disappointing, especially in front of our home crowd."
