A brace of penalties from the boot of fly-half Ian Keatley handed Connacht a 6-3 lead at half-time in a closely fought contest at wind-swept Galway.
But Bibo burst through on 57 minutes before wing Carr followed nine minutes later to settle the tie.
Keatley converted both before Mamuka Gorgodze touched down in the final minute for the French team. However their failure to gain a losing bonus point meant Connacht cannot be passed for top spot in the group.
Connacht travel to Madrid next week bidding to clinch top seeding and with it a guaranteed home draw in the quarter-finals.
Final Score Connacht Rugby 20 (3) Montpellier 10 (3)
Scorers
Connacht Rugby
Try - Fionn Carr, Liam Bibo
Conversion - Ian Keatley (2)
Penalty - Ian Keatley (2)
Montpellier
Try - Mamuka Gorgodze
Conversion - Maxime Castell
Penalty - Adrien Tomas
Attendance: 1,700
Referee: Peter Allan (Scotland)
Teams
Connacht Rugby: T Nathan, L Bibo, N Ta'auso, K Mathews, F Carr, I Keatley, F Murphy, B Wilkinson, S Cronin, R Morris, M Swift, B Upton, M McCarthy, R Ofisa, J Muldoon [capt]
Replacements: A Flavin, R Loughney, J Hagan, A Browne, G Naoupu, C O'Loughlin, M Nikora, A Wynne
Montpellier: J Alcalde, B Thiery, O Smith, G Rees, T Brana, J Schutte, A Tomas, E Van Staden, J Caudullo [capt], B Douglas, M Macurdy, M Gorgodze, V Bost, G Chkhaidze, M Giraud
Replacements: F Rofes, N Leleimalefaga, M Demarco, S Matadigo, M Castell, G Doumayrou, S Max, J Figallo
