Downing fired home with his right foot during the opening minutes of the second half to give the Reds the three points at Anfield.
"Stewart Downing scored a brilliant goal," said Rodgers. "He is a wonderful technician off either foot.
"He has had a rough ride here, it's been a tough time, but he is a good guy and an important member of our team and he got a good reception from the Kop."
The win has earned Liverpool a two-point lead at the top of Group A after three matches.