The Reds won 1-0 thanks to Stewart Downing's effort.
Reflecting on the performance, which yielded a third straight clean sheet in all competitions, Rodgers said: "Anzhi have a terrific group of players - they have invested heavily in it and they have a top manager.
"So to get a win and keep another clean sheet was excellent, and I thought we deserved it.
"I thought our intent in the game was good. We wanted to go forward, and particularly in the second half, we stepped it up and did very well.
"And it was a wonderful strike by Stewart - a great goal, and a good one to win the game."
Rodgers fielded a strong side including the likes of Steven Gerrard and Luis Suarez, resisting the temptation to rest players ahead of Sunday's Barclays Premier League derby at Everton and making only three changes from the win over Reading on Saturday.
Regarding his selection, Rodgers said: "For us, it is about taking it one game at a time.
"Our squad is not overly big, but we were at home and the condition of the players physically is very good. So I had no concerns about that.
"The work-rate in training and the games has been fantastic. We will recover well after a very good performance and result, and now we are all really looking forward to Sunday."