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There were no tries in the first half, but Ben Botica was able to give the home side a narrow lead at half time, kicking three penalties in the opening 40 minutes.
Danny Cipriani had just one attempt at the uprights in the first half, but he made it, scoring what would be the visitors' only points of the match.
Quins were able to extend their lead early in the second half when Luke Wallace crossed the whitewash to score the first try of the match.
Sale defended well as the hosts pushed for another try to seal the win, but the visitors let Jack Clifford through just over 10 minutes from full time.