Yes, and if you want to swap it around (My birthday is today), that's fine too although that will have a different focus. Your sentence focuses on why today is special; my other sentence focuses on when your birthday is.
"It's my birthday today" is more colloquial. I can't imagine myself saying "Today is my birthday". I could of course use the contracted form "Today's my birthday", but if I prefer to use "It's my birthday today" it doesn't cost me anything.
I don't really understand the concept of "wordy" when we're talking about everyday expressions in spoken English .