Iceland, ranked 115 places below their hosts at 131, shocked Les Bleus by taking a two-goal lead through Birkir Bjarnason and Kolbeinn Sigthorsson.
Laurent Blanc's side reduced the arrears through Mathieu Debuchy seven minutes after the break but they had to wait until the 85th minute to equalise through Franck Ribery before Adil Rami's late strike won it.
It was a tougher examination than France would have expected to face against the Atlantic minnows but Evra told L'Equipe: "We must not be too hard on ourselves.
"After one match, you cannot know exactly what you need to do to improve. The second match is different, the third as well.
"You can play very well in your warm-up games and lose the first match at the Euros. It is June 11 we must be ready for."
Evra appeared culpable for Sigthorsson's goal when he was beaten by winger Rurik Gislason, who crossed for the Ajax striker to finish.
But the Manchester United full-back said: "When we concede goals, it is down to everyone. We should not single out the defence, the attack or the midfield."
Fellow full-back Debuchy may have got forward to score his first international goal, but the Lille man admitted his side's defensive display was not up to scratch.
"We were poor, it is a shame," he said.
"In the first half, they had two chances and they put them away. We had a few chances but we did not put them away."
Les Bleus face Serbia on Thursday and Estonia next Tuesday before heading to the championship, where they are in Group D with England, Sweden and co-hosts Ukraine.