Akinfeev somehow failed to keep out Lee Keun-ho's speculative effort, gifting South Korea a 68th-minute lead in Tuesday's game.
Russia did find a response through Aleksandr Kerzhakov, yet will have been disappointed to only collect a point.
"I want to apologise,” said Akinfeev. "It was a ridiculous goal. Any normal person should apologise and blame themselves for this moment.
"The goalkeeper of the national team should not give away such goals. The coach and the players have supported me."
Russia will still be hoping to reach the next phase, with Algeria and Belgium completing Group H.