The Super Eagles were beaten 2-0 on Monday thanks to two late goals in Brasilia.
Despite Nigeria's exit from the tournament, Mikel believes that the side performed well against the French.
"Honestly, I don't think that we should be feeling bad right now," the midfielder told FIFA.com. "We've had a great tournament and we showed the whole world just what we're capable of, even when up against a great team.
"I think that we played really well from the start, right up until we conceded the opening goal. From that moment on, I'll admit that we let our heads drop a bit.
"We couldn't drag ourselves back into it, but none of that should take the shine off our performance."
France will now face Germany in the quarter-finals.