The loss was Montreal's fourth straight and its fifth overtime loss this season, most in the NHL.
Both Brian Gionta and David Desharnais kept the shootout going for Montreal with goals in the third and fourth rounds, but when Pavelski tallied in the sixth round P.K. Subban could not answer for the Canadiens.
Each team scored a goal in each period in regulation, with Montreal scoring first on each occasion.
Clowe's goal came with 1:16 left in regulation.
Antti Niemi made 29 saves for San Jose before letting in two goals in the shootout.
The loss went to Carey Price, who stopped 38 shots.