An unfortunate own goal from goalkeeper Cedric Carasso was the difference between the sides, as both sides struggled to create opportunities at the Estadio da Luz
The home side were the team in form heading into the clash, after losing just once in 2013, while Bordeaux had lost their last four league games.
However their form in Europe has been impressive. They beat Dynamo Kiev in the last round and are unbeaten in six European games.
Benfica are duelling with Porto for the Liga title, with the Lisbon side currently ahead by two points.
After a slow start the hosts took the lead after 21 minutes when Moreno Rodrigo's 25-yard effort cannoned down off the crossbar onto the back of Carasso and into the net.
Two minutes later Carasso made a more positive contribution to parry away Rodrigo's effort.
Bordeaux tried to force they way into the match and the home side needed Luisao's last-gasp challenge to deny Diego Rolan who was closing in on goal.
Bordeaux lost Julien Faubert just before the break through injury, but they ended the half on top, with Artur Moraes called into action to deflect Ludovic Obraniak's 20-yard effort away to safety.
Three minutes after the restart, Oscar Cardozo's half volley into the ground was pushed to safety by Carasso.
After a slow start to the second period, Nicolas Gaitan released Lorenzo Melgareja down the right flank but his cross was just too far in front of Rodrigo and substitute Eduardo Salvio.
Artur Moraes denied Fahid Ben Khalfallah late on for the Portuguese side, who will be confident of progressing in next week's second leg at the Stade Chaban Delmas.