The visitors relied heavily on penalties to build a head of steam, before holding on at the end to seal their passage through to the last four of the competition.
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Benoit Paillaugue made no mistake from the follow-up attempt, while also adding three more points from a pen just two minutes after that breakthrough at the AJ Bell Stadium.
It was to be Sale who took the lead into the interval thanks to winger Tom Brady, who crossed over in the final 10 minutes of the half to keep the scoreline very narrow at the midway stage.
Paillaugue added five successful penalty attempts to his tally in the second 40, though, either side of Cipriani's own three-point haul, to put the Top 14 side in control.
That proved to be too little too late in the end, as Montpellier saw out the remaining time to end their opponents' unbeaten home record and progress through in the process.