Although I don't have the book here, I am quite certain that mixed strategies are already present in the book from 1944. I believe that I have read somewhere the claim that "mixed strategies" where actually introduced in an earlier version of the book (published in German in 1928, says Wikipedia). However, what I have read mostly is the claim that Emile Borel (yes, THE Borel) first wrote down this idea in a note in 1921.
I guess here is a man with enough authority to state that:
E. Borel was the first author to evolve the concept of a strategy, pure as well as mixed, although he did not go beyond the case of the symmetric two-person game.
says von Neumann here: "Communication on the Borel Notes", J. von Neumann and M. Fréchet, Econometrica 1953.