There is a simple proof that a game of Hex must have a winner, which implies the result you want.
Take a look at [The Game of Hex and the Brouwer Fixed-Point Theorem][1] (by David Gale. The American Mathematical Monthly, Vol. 86, No. 10. (Dec., 1979), pp. 818-827). The Brouwer fixed point theorem and the Jordan Curve theorem follow from this.

Also, there is a short proof here: [Brouwer's Fixed Point Theorem and the Jordan Curve Theorem][2].


  [1]: http://www.math.pitt.edu/~gartside/hex_Browuer.pdf
  [2]: http://www.math.auckland.ac.nz/class750/section5.pdf