As you suggest, the fact that $e^x$ is increasing is a useful property here.  In particular, that lets you in some cases to apply Markov's/Chebychev's inequality to the random variable $e^{tX}$ in order to get exponentially decaying bounds on the tails of $X$; see e.g.
[Chernoff bounds][1].  In principle any other positive increasing function could be used in the same way, but $e^x$ is a particularly useful choice because it is especially well suited to studying sums of independent random variables, as noted already in Yuval's answer.

  [1]: http://en.wikipedia.org/wiki/Chernoff_bound