So far I've been doing AABB-Plane intersection tests to find the collision between an AABB and a mesh. This works fine if the AABB collides with one of the sides, however if it hits one of the edges I can't use this method:

(The AABB starts at the top right and moves to the bottom left, red and green are the intersections with the mesh planes)

I'm at a loss, could anyone point me in the right direction?

You can use the separating axis test for this. I can a presentation this year at the GDC. The presentation is quite dense so I also decided to release the source code. You can find it here:

https://box2d.googlecode.com/files/DGregorius_GDC2013.zip