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 (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.