My Arab mare puffs up occasionally too. Now I usually tighten her cinch 3 different times. First, just snug it up so that the saddle can't slip off if she moves around. I'll walk away, give her a minute then come back and tighten it up a little bit more. After I tighten her up the second time though, I'll run my fingers under the cinch and pull it away from her a little (on both sides) and slide my fingers up and down to make sure there is no pinching going on. I usually do this though if it's winter and her coat is long or if I'm using a rope cinch (which I don't use anymore). After that, I'll hand walk her a couple laps, then finish tightening it up.
I ALMOST had that happen to me when I was really young 5-6? I was riding my Shetland Pony and she wound up taking off galloping being a butt head (she was good at that). As she was galloping though, my saddle started to slip off to the side. I am lucky it didn't slide totally around like you though! It's quite a scary thing to go through even without falling off! If you go through these steps that people have mentioned above, I'm sure you can avoid that next time!