Sale scored early and late tries but a dominant 20-minute period saw Montpellier win their Pool Six clash.
Mark Cueto crossed after three minutes following a superb Danny Cipriani chip, and Richie Vernon added a try late on.
But in between, the French side's pack - and some excellent play by fly-half Francois Trinh-Duc - created three tries in the space of eight minutes.
Sale take no points from the game, but go into a double header with Toulon in second place in a very difficult pool.
Cipriani demonstrated his famous qualities of invention when, at the end of a swift counter-attack prompted by a crucial turnover, he chipped a perfect kick behind the covering defence for Cueto to collect on the bounce and score in the corner.
It was the perfect start for both Sale and Cipriani but their advantage was soon reduced by three points when Martin Bustos Moyana kicked a penalty and then the wing put the hosts in front with his second shot at goal eight mintues later.
Bustos Moyana slotted a third penalty as the visitors infringed again but Sale closed the gap again as Rob Miller kept them in range with a calm penalty of his own.
The French team's pack first showed its teeth on the half-hour mark, crumbling the opposing forwards at a scrum and although the penalty was missed, it was to prove an ominous sign for Sale.
Over-zealous counter-rucking allowed Miller to reduce the deficit to a single point early in the second half, but Montpellier reasserted themselves quickly.
They consistently looked to utilise their fearsome all-Argentine front row to establish position at the set-piece and, from a scrum, fly-half Trinh-Duc accepted a pass and sped through the remaining defence to score from 10 metres.
It was one of the rare occasions that - in the difficult conditions - the ball stayed in hand for either side but the French side's front five were fully in control as the game wore on.
Trinh-Duc's clever chip over the first line of defence was the catalyst for a counter-attack that ended with a try for Thomas Combezou in the corner and, with the floodgates momentarily open, the stand-off created an exceptional third try for Julien Tomas.
A string of substitutions put an end to the hosts' momentum and allowed replacement flank forward Vernon to capitalise on an error and score a try that Miller did well to convert.
But there was to be no further scoring for the visitors, who now face a December double-header against one of the favourites for the trophy this season, Toulon.
Sale head coach Bryan Redpath:
"Winning here is a difficult challenge and winning matches in Europe is hard. We just didn't have enough firepower to compete.
"It got a bit messy and sloppy for us, and we must give credit to Francois Trinh Duc. He set up two great tries with chip kicks and finished one himself.
"We could have crumbled at 12-8 but didn't. We fought back, we had opportunities, but they got a couple of lucky bounces.
"This competition is for the very big clubs and it's hard to compete with them at the moment. The Premiership will be and always has been our main focus.
"We wanted to make sure that the players didn't embarrass themselves and I don't think they did at all."
Montpellier: Amorosino; Berard, Nagusa, Fernandez, Bustos Moyana; Trinh-Duc, Tomas; Figallo, Creevy, Bustos, Fakate, Privat, Ouedraogo, Gorgodze, Beattie.
Replacements: Combezou for Berard (41), Bias for Gorgodze (50), van Vuuren for Creevy (50), De Marco for Fakate (63), Watremez for Bustos Moyano (64), Bosch for Trinh-Duc (65), Paillaugue for Tomas (67)
Not used: Pelo
Sale Sharks: Miller; Brady, Addison, Jennings, Cueto; Cipriani, Willis; Roberts, Ward, Buckley, McKenzie, Myall, Gaskell, Seymour, Easter.
Replacements: Taylor for Ward (41), Leota for Roberts (54), Harrison for Cueto (54), Cobilas for Buckley (55), Vernon for Gaskell (58), Fowles for Willis (61), MacLeod for Jennings (71)
Not used: Doyle
Sin bin: None
Referee: Neil Paterson (Scotland)