However, the only goal came in the 27th minute, Benzema converting a Marcelo cross at the back post.
Liverpool are now three points behind second-placed Basel, who crushed Ludogorets Razgrad 4-0 in Switzerland.
Dortmund followed up a 4-0 win away to Galatasaray in Turkey two weeks ago by defeating the same team 4-1 in the return fixture.
Arsenal should also have clinched their place in the last 16 on Tuesday as a Mikel Arteta penalty and strikes by Alexis Sanchez and Alex Oxlade-Chamberlain put them 3-0 up at home to Anderlecht.
However, the Belgian champions produced a remarkable comeback to draw, Anthony Vanden Borre scoring twice before Aleksandar Mitrovic equalised in the last minute to keep them in the competition.
In Group A, Atletico closed in on qualification with a 2-0 win over Malmo in Sweden.
Meanwhile, in Turin, Juventus came from behind to beat Olympiakos 3-2 to give their own hopes of going through a shot in the arm.
Leverkusen are in control of Group C thanks to an impressive 2-1 triumph against Zenit St Petersburg in Russia.
And Portuguese champions Benfica revived their European campaign by defeating Monaco 1-0 in Lisbon with a late Talisca strike.