Former Liverpool striker David Ngog gave the visitors a great start when he netted after just five minutes.
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Aissa Mandi doubled their advantage on the stroke of half time to put Reims firmly in the driving seat against their fellow strugglers.
However, Giovanni Sio struck for Bastia just two minutes into the second period with a smart finish.
The visitors held on to take the points, and the defeat leaves Bastia two points above Lorient, who currently occupy the final spot in the relegation zone in France's top division.
The victory was Olivier Guegan's first since taking over as Reims boss earlier this month.