As thundara demonstrated, the problem as stated is quite straightforward: As long as you're within distance r/4 of the center (r being the radius), you can remain on the opposite side of the center from the monster. That means that you have only (3/4)r distance to row straight out, while the monster must run halfway around the lake, a distance of πr. Since πr/4 > (3/4)r, you can simply row straight out from that point and you'll win. In fact, as long as the monster is slower than π + 1≈ 4.14 times as fast as you, you can live without having to make any complicated maneuvers.

A more interesting question is: What is the worst case? That is, how fast can the monster run and still be unable to catch you? Seems like we will need curves for this one. Not sure if you need the calculus of variations or if there's a more elementary approach.

Update: There's a link on stackexchange to an article at generic maths. This article claims that a straight path is best, although there is a better straight path than the one I gave:

if you rowed on a curved line, you'd end up at a location that you could have gotten to quicker with a straight shot.

Seems reasonable, but not entirely convincing. Update 2: I'm convinced now.