Sam Gilbert kicked an early penalty to put Connacht ahead on three minutes before Montpellier hit back when they were awarded a penalty try three minutes later following a five-metre maul.
Things went from bad to worse to Connacht when they were reduced to 13 men for a spell as both Dylan Tierney-Martin and Shamus Hurley-Langton were shown yellow cards.
Montpellier took full advantage and added a second try on 14 minutes through Christopher Tolofua.
Returning to full complement, Connacht battled back and reduced the deficit when Tierney-Martin finished off a free-flowing move in the corner.
Mohamed Haouas and Domingo Miotti then crossed before the break with Shayne Bolton also shown a yellow card for Connacht, as Montpellier went in with a comfortable half-time lead.
Sean Naughton then cut the gap for Lancaster’s side soon after the restart as he crossed under the posts, but again Montpellier responded to re-establish their healthy advantage.
Auguste Cadot scored the pick of the tries for the hosts on 52 minutes as he weaved his way through a number of challenges before racing over with Ali Price adding a sixth by diving over the line from close range to all but secure the win.
Jack Aungier scored a late consolation try for Connacht before Valentin Welsch rounded off the scoring with a last-minute score for Montpellier.
Montpellier: Banks; Taofifenua, Cadot, Anyanwu, Echegaray; Miotti, Price; Erdocio, Tolofua, Haouas, Verhaeghe, DuGuid; Camara, Becognee, Vunipola (capt)
Replacements: Akrab, Forletta, Welsch, Tauleigne, Nouchi, Rates, T Vincent, A Vincent
Connacht: Gilbert; Bolton, Forde, Aki, Mullins; Naughton, M Devine; Bohan, Tierney-Martin, Illo; Joyce, Murray; Prendergast (c), Hurley-Langton, Boyle
Replacements: Victory, Dooley, Aungier, J Murphy, O’Brien, B Murphy, Carty, J Devine