You have two choices when playing from the fringe: Keep the ball on the ground, or put it in the air. My advice is to always roll it using a putter, a shot called the Texas wedge.
This play takes a lot of guesswork out of the equation. You don't have to puzzle over how far to fly the ball or worry about how it will react when it lands on the green. Also, the putting stroke has the fewest moving parts of any swing, so unlike with a wedge, you probably won't skull, shank or chili-dip the shot.
To execute the Texas wedge, take your standard putting setup (left). Then lean a little forward [toward the target] without moving your feet or altering the ball position (right). The closer the ball is to where the fringe meets the rough, the more you lean forward. This will naturally help produce a more up-and-down stroke, which keeps the putter from getting caught in the thicker grass of the fringe and makes it more likely you'll make solid contact.