The Lithuanian club improved to 5-3 in Group D with the triumph while Montpellier fell to 1-7.
USA international Crystal Langhorne had 23 points and nine rebounds and Jurgita Streimikyte weighed in with 14 and 12 for TEO while Sheana Mosch scored 18 in defeat.
Lattes Montpellier trailed 28-18 early in the second quarter but fought back to tie the game, with Khadijah Whittington's jump shot making it 28-28.
Over the remaining 4:14 of the opening half, the contest stayed tight and TEO went to the changing rooms with a 35-34 lead.
The home side limited Montpellier to just four points for the first six minutes of the third quarter and that proved to be the visitors' undoing.
They were outscored 24-15 in the frame and got no closer than eight points over the last 10 minutes.
Rutenis Paulauskas, coach TEO Vilnius:
"We played well in defence in the second half. We also successfully used our big players and scored many points in the paint. We still have to improve and not to make so many turnovers."
Valery Demory, coach Lattes Montpellier:
"It was a hard game. Everything was good while we could score on the fast break. After that I let my players to rest since the main aim for us this season is the French championship and we have a tough game ahead of us this weekend."