This is a particular case of the following well-known statement in algebraic geometry. 

> **Proposition.** Let $k$ be an an algebraically closed field of characteristic zero and  $f \colon X \to Y$ a morphism between integral $k$-schemes of finite type over $k$. Then if $f$ is bijective and $Y$ normal the morphism $f$ is an isomorphism, that is $f^{-1} \colon Y \to X$ is a morphism as well.

In your situation, $k= \mathbb{C}$ and $Y$ is smooth (hence normal), because it is an open dense subset of $\mathbb{C}^2$. So the previous result applies. 

See also the following MathOverflow question:
http://mathoverflow.net/questions/73321/isomorphism-between-varieties-of-char-0