How am I winning a round?

The concept of winning isn't based on health. Basically, you keep hitting people until you fill up the bar on the bottom of your screen that is next to your character's picture. When that bar reaches level 1, you can either use the super (L2) to kill someone to earn a point, or keep smacking people around to build up to levels 2 or 3. The main purpose of the game is to get the most kills from your supers as possible. If course, there is also different modes for fighting, like Stock where there are limited lives for you and your opponents. But you still need to kill them with your supers in order to win.