City won 2-1 on the night for a 4-2 aggregate victory to reach the quarter-finals where they will face Lyon.
They capitalised on two shocking mistakes by Varane to get their goals, with Raheem Sterling and Gabriel Jesus scoring either side of a Karim Benzema strike for the visitors in this last 16 second leg, played at an empty Etihad Stadium.
Varane was dispossessed inside the box in the ninth minute allowing Sterling to put City ahead, and his poor headed backpass attempt gifted Jesus another goal midway through the second half.
"I want to stand up because in my eyes, this defeat is on me. I have to take responsibility for it," admitted Varane.
"I am sad for my team-mates, for their efforts. I have to take my responsibility in the defeat, mistakes cost dearly at this level of competition."
The 27-year-old French national captain added: "We gave everything on the pitch but sometimes you can fail. Today, I failed, and I have to admit it."