How does element dominance work in Avatar?
It works a bit differently in Avatar.

Firebenders draw they power from great sources of fire. The sun is usually the strongest one around (with the exception of Sozin's comet, when it arrives every 100 years). Water benders draw their power from the moon.

Therefore, at night, the water bender will usually win, while at day the fire bender will usually win. Since the fire benders were on the offense, they get to decide when to attack (during daylight).

Also, there seem to have been a lot more fire benders than there were water benders (also, the water benders were separated into two very faraway clans). 

Last but not least, in Avatar, the relationship between the elements doesn't seem to be that of dominance over one another, but rather, Earth and Air are opposites, while Fire and Water are opposites. It's not that one is stronger than the other, but rather two sides of the same coin.