Jonjo Shelvey spurned the best chance of a tight first half when the England international miscued his shot with only the Anzhi goalkeeper to beat.
Downing was switched to left-back after half-time with the introduction of Raheem Sterling, and the former Aston Villa winger cut in from the wing to fire past Vladimir Gabulov in the 53rd minute.
“It was a brilliant goal,” said Rodgers.
“We asked him to cut inside from left-back. He’s a wonderful technician Stewart – he’s got great technique off either foot, his strike was wonderful.
“He’s had a rough ride here really and it’s been a difficult time for him, but he got a great round of applause from the Kop, and if you can get that from the Kop, it will give you a massive lift going forward.
“He’s a good guy and he’s working very hard and he’s a very important member of our team.”
Rodgers’ men are now top of Group A after Young Boys beat Udinese, and the Liverpool manager revealed Sunday’s Merseyside derby never played a part in his decision to field an almost full-strength side against the Russians.
“It was great credit to the players. To be fair, I haven’t been thinking about the derby,” he added.
“For us, it’s about taking one game at a time at the moment and Anzhi are a very good side with some outstanding players.
“But tonight was about the team and the group, some of the guys who have played in early rounds have been brilliant for us.
“I’ll always pick a team who I think can win the game and tonight it was an excellent performance.”
MORE: Man United latest news
MORE: Arsenal latest news
MORE: Chelsea latest news
MORE: Liverpool latest news