Fursenko, who also sits on the UEFA executive board, resigned after meeting Russian President Vladimir Putin.
"I would like to apologise to our fans for such a result. I have taken a difficult decision - to step down as head of the Russian Football Union (RFU)," Fursenko told his close friend Putin, the Kremlin said.
Fursenko has been heavily criticised for the team's poor showing at the tournament in Poland and Ukraine, where they failed to advance past the group stage following a 1-0 upset by unfancied Greece in their final Group A match.
Fursenko, 58, was appointed as the RFU boss in February 2010 after working as the president of Russian Premier League club Zenit St Petesburg for four years.
He was largely responsible for hiring Dutchman Dick Advocaat as Russia coach in 2010 after deciding against extending a contract of fellow-Dutchman Guus Hiddink, who led the Russians to the Euro 2008 semi-finals. The team then failed to qualify for the 2010 World Cup.
Fursenko's resignation was greeted with mixed reviews from Russian football experts.
Former Prime Minister Sergei Stepashin, who now serves as chairman of Dynamo Moscow's advisory board, said: "Fursenko's exit was justified taking into account our result at Euro 2012."
However, honorary RFU chief Vyacheslav Koloskov criticised Fursenko for abandoning "the sinking ship".
"I have a feeling there is more to his resignation," long-time RFU boss and former FIFA vice-president Koloskov was quoted as saying by local media.
"Just three or four days ago Fursenko said he was looking for a new coach, meaning he had no plans to resign. Something must have happened that made him change his mind," Koloskov said.
"I'm not going to condemn him - I was in his position myself a few years ago (after Russia's fiasco at Euro 2004) - but I think it's not right for him to abandon the sinking ship."