Valid claims for a draw in a blitz game The following situation occurred in my last tournament.
It was a 5 min blitz game without time increment. My opponent didn't have any material left. I touched my rook and wanted to place it on the back rank to deliver checkmate on this move. While I am moved my rook, my opponent claimed a draw because my flag on the chess clock has fallen Nevertheless, I moved my rook and announced a checkmate. Then I saw both flags have fallen. Then the arbiter arrived at the board and gave me the point because he said he sees a checkmate.
Is it valid for my opponent to claim a draw, even if it's not his move?
What would be the correct way for my opponent to claim a draw?
Well, let's look at the FIDE rules:

Yes, your opponent may claim a draw even if it's not his move. He should then stop the clock (rule 6.12b):


  A player may stop the chessclock only in order to seek the arbiter’s assistance


but, of course, it will be his word against yours that he did that before you released your rook, because checkmate immediately ends the game (rule 5.1).

I think it is up to the arbiter to decide which happened first, but as your opponent didn't stop the clock to claim the draw, I'd be inclined to the same decision: to award you the full point.