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The home side were ahead just two minutes into the match when Montpellier were caught out by a pass out wide to Banahan, who was left to stroll in for his first try of the night.
The wing was over again within the opening quarter of an hour when a dummy from Sam Burgess left Banahan open for another easy score.
After Semesa Rokoduguni knocked on to deny himself the chance to increase Bath's lead early in the first half, the hosts sent on some fresh legs, and they quickly set up Banahan's third of the match.
Montpellier finally got off the mark in the final 20 minutes when they won a lineout just five metres out, and were able to make the most of the opportunity as Alex Tulou powered over.
Mickael Ivaldi added another late on for the visitors before Bath pushed for a fourth try to secure a bonus point, and they clinched it with just over a minute to spare when Horacio Agulla found a gap.
The result moves Bath up to second in Pool Four, while Montpellier remain bottom.