The Russians have been heavily criticised by fans, media and soccer experts at home - with Arshavin viewed as the main culprit - after they failed to advance past the group stage in Poland and Ukraine.
"The result has disappointed us and, of course, al our fans. We understand that," Arshavin told a local TV station in St Petersburg on Thursday, five days after Russia's exit following a 1-0 upset by unfancied Greece in their final Group A match.
"As team captain, on behalf of our players and myself personally, I would like to apologise for our performance and the result that we had.
"It's not right to shy away (from criticism). That's why I took this moment to say something on this subject. I will do it only once and will not talk about this any further," he added.
Arshavin, who played well in their opening 4-1 win over the Czech Republic, was involved in a much-publicised spat with Russian fans in the team's hotel in the centre of Warsaw just a few hours after their last match.
After one of the fans, a member of the Russian Parliament, demanded an explanation for the team's poor showing, Arshavin said: "If you have had high expectations for our team here, then it's your problem, not ours."
Arshavin's words sparked a storm of negative publicity in Russia.
"How can a citizen of Russia treat the honour of his country like that?" Valery Golubev, deputy chairman of Russian energy giant Gazprom, was quoted as saying by local media.
"He should be deported."
Gazprom is the owner of Russian champions Zenit St Petersburg, where Arshavin made his name as a player before joining English Premier League club Arsenal in 2009.
Arshavin, widely considered Russia's most talented and best-paid player, is also facing an uncertain club future.
He was loaned to Zenit last season following a turbulent year at the London club.
Zenit have yet to express their interest in re-signing the 31-year-old forward permanently.