Hit it FAST. Faster than you think you can. The berm will help you out A LOT. I'm also assuming you're riding flats (because you mentioned putting a foot down when you felt like you were going to fall). If you have that safety net just get a good run up and hit it. You'll feel it when you stick it.

Practice is probably the best thing. If you practice on a pump track or somewhere like that though be careful when you actually get out on the trail. Pay attention to where your front tire is going if the berm is going downhill. You don't want to fly off the top because the berm drops out from underneath you.

The only problem with trying to keep your body upright like you mentioned is that it sounds like that would lead to more problems feeling like you're leaning too far. If you're upright you'll be pushing down on the bike from the side almost trying to push it over, you want to be in line with the berm and pushing INTO the berm. Keep your head upright to watch where you're going but make sure that at least most of your force is going straight into the surface of the berm, not down on the handlebars from the side. Hope that helps... good luck.