I did a quick search for how to reduce raveling when cutting burlap and found great stuff on cutting straight lines, but nothing about cutting on the diagonal. My quick search lasted much longer than I wanted it to, and I finally gave up on someone else doing the trial and error for me.
I decided to try painting the burlap with Modge Podge. After trying to put it on straight out of the bottle, I realized it would go on much better if I thinned it down a bit. I coated the whole piece and waited for it to dry.