The Chelsea midfielder claimed referee Mark Geiger saw Giroud strike him in a tangle around the half hour mark, but opted against showing the Arsenal man a card before France battled to victory with two late goals to secure a place in the quarter-finals.
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"Yes I got an elbow and the referee saw it and he only gave him a warning, not even a yellow card, nothing," Mikel told Mirror Sport. "It was bad.
"We know the game - if you see it, act on it. The ref saw it. He did catch me, I don’t know if it was too hard but the intention was definitely there.
"I don't think Giroud is a dirty player, I have played against him several times before and everyone wants to win for his country. He hasn't done those sort of things before so maybe it was just one of those things in the heat of the moment."
It was an even more painful match for Mikel's midfield colleague Ogenyi Onazi, who left on a stretcher during the second half following a challenge with Blaise Matuidi.
Mikel reported a suspected broken ankle for Onazi and, although he refused to criticise Matuidi, he again questioned Geiger.
"I thought the referee didn't help us at all. There were some heavy challenges and I think Ogenyi Onazi has broken his ankle. Challenges like this are bad but it was a 50-50, he just miscalculated," Mikel said.