"The fans were right behind us and the atmosphere was unbelievable," said former City man Sinclair.
"That's what spurred us on. We kept going, going forward and creating chances.
"We keep improving. We learned from the (7-0) Barcelona defeat. We took the negatives out of it and performed tonight."
Brendan Rodgers' side now sit third on the Group C table on one point, five behind leaders Barcelona and three in arrears of second-placed City.