Gignac, who also scored in Sunday's 1-0 win at Nice, grabbed his 11th league goal of the season when he beat visiting goalkeeper Cedric Carrasso in the 41st minute.
He needed a large dose of luck though as Bordeaux centre-back Lamine Sane slipped in the box while trying to intercept a long pass by Benoit Cheyrou.
"I appreciate how solid our team are and the fact they never give up," coach Elie Baup told reporters after his side stretched their unbeaten sequence to five games.
Bordeaux, who have scored only three goals in eight league games, are ninth with 43 points.
"Unfortunately it's always the same, one game after the other. We had the possession but we were no harm to them," said Bordeaux midfielder Ludovic Obraniak.
The hosts went close after 10 minutes when Foued Kadir took advantage of confusion in the box only to see his curling effort go just wide.
The fare was generally dull in the first half as Marseille toiled without playmaker Mathieu Valbuena.
The second period was not much better with home keeper Steve Mandanda making his first save in the 70th minute by parrying a Nicolas Maurice-Belay shot.
Henri Saivet also headed over the Marseille crossbar in the 75th minute.
Marseille's clean sheet was their fourth in succession in the league.
