There is an applications section in iTunes.
You should see an Applications Tab when you plug in your iPhone with 2.0. You might have to check the option to sync them to your phone and then click apply.
FWIW, I downloaded 2 games on iTunes, sync'd them, downloaded a 3rd on iTunes and downloaded another directly to the iPhone and iTunes sync'd up everything beautifully.
This was a gen 1 iPhone with 2.0 software.