Another thing is to avoid using abilities while you're in the red, and if you can, use an ability as soon as you're out of it. It seems the client sends your position every time you use an ability, so if you use one while you're in the red, that might be the last update the server gets before the AOE goes off.

Basically, though, just move ASAP. Don't ever try to guess when you need to move to make it on time. You need to move now.

And learn the mobs. If their worst AOE is a cone, stick as close to them as you can, so you don't have as far to run (forward and/or to the side) to get out. If it's a circle around them, stay at max melee range for the same reason. If it's a line, whatever you have an easier time with. You can also develop a bit of a sense for when they're going to be using it; a lot of mobs are very predictable with their AOEs, so you can be (more) ready to move in the second or so beforehand.

And another thing! For some moves, the markers simply aren't big enough, or possibly shaped wrong. The 50 WAR class quest fight, for example, has one move with red markers that are very clearly not representative of the actual effect (couldn't tell if it showed a line but was actually a cone, or was just a much wider line than displayed). Don't stop just outside the red unless you're sure it's actually safe. Most of them are right, but there are a few that aren't, unfortunately, and all you can do is learn the bad ones.