It was a game between Texas A&M vs. the University of Texas, but the "broadcast" wasn't exactly one you'd recognize. It was done in Morse code from the site of the game and relayed back to Texas A&M, where the "action" would be translated and then spoken on a horn mounted out a window to a waiting audience. 

Hardly the kind of first you'd expect. (A previous attempt a year earlier failed.) And while the anniversary is getting a lot of attention in Texas, it may not even be true. KDKA Pittsburgh claims it broadcast a Pittsburgh Pirates baseball game to some shortwave listeners on August 5, 1921 - predating the Aggies claim by three months. Whoever it was, the business of broadcasting sports has come a very long way since those early days. 

And by the way, if that Texas game WAS the first broadcast, it was anti-climactic - it ended in a scoreless tie.  

Celebrating the 100th anniversary of the first radio play-by-play broadcast