Yep. And just for clarification, think of the .dmg as a discardable disk. You have to load it up (you see the drive on the desktop) and you can access the files on it.
Once you load it all up, just delete it (same as throwing the disk away) and you're good to go. The files themselves are copied (well should be copied) to the computer.
I was pretty confused when I first got my mac and just ran the applications off of the .dmg as well.. then I found out it's like a virtual "disk"; that's why it's mounted.