The Cameroon international joined the Russian outfit from Inter Milan in the summer and has kicked of his career with two goals from three league appearances.
After the weekend's loss at Kuban, Eto'o said that the traveling fans made life so much easier and likened them to the supporters at Anfield after he played there a few times with Barcelona.
"Of course I felt their support," Eto'o told Anzhi's official website.
"My team-mates and I personally are very thankful to them. Owing to our supporters we never feel lonely in away matches.
"They're like Liverpool fans with their famous hymn, 'You'll Never Walk Alone'."