A 1-0 defeat to Greece followed a draw with Poland to KO Russia - despite an impressive 4-1 win over Czech Republic in their opening game.
Arshavin said: "The result has disappointed us and, of course, all our fans. We understand that.
"As team captain, on behalf of our players and myself, I would like to apologise for our performance and the result that we had.
"It's not right to shy away. That's why I'm saying something. I'll do it only once and will not talk about this any further."