Jean-Luc Vasseur's men had clinched just seven points prior to the clash at the Stade Auguste Delaune and Reims' dismal start will have been widely expected to continue against a Bordeaux team looking to overtake Marseille at the summit of the French top flight.
But Fortes' 32nd-minute strike gave the hosts all three points, although there was a large degree of fortune about the goal.
The Cape Verde winger worked his way down the right before delivering a cross that took a deflection and swerved past a stranded Cedric Carrasso in the Bordeaux goal.
Willy Sagnol's men pressed hard for a leveller and looked to have found one in the 59th minute when Mali striker Cheick Diabate slotted home, only for referee Frank Schneider to chalk off the goal after Mohamed Fofana's attempted clearance had struck the hand of Diego Rolan in the build-up.
Substitute forward Emiliano Sala then wasted a glorious chance to restore parity in the 79th minute as he headed wide with Reims goalkeeper Kossi Agassa.
Bordeaux continued to dictate play in the latter stages, but their search for a point proved in vain as Reims held on to earn a victory that moves the Champagne-Ardenne club out of the bottom three.
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