Well, the Dallas Eagles opened their 1950 season with a baseball game in the Cotton Bowl. As the story goes, the Eagles sent out an all-star lineup including Home Run Baker, Duffy Lewis, Charlie Grimm, Tris Speaker, Charlie Gehringer, Travis Jackson, Ty Cobb, Mickey Cochrane and Dizzy Dean. These old-timers took the field ceremonially for one batter and then gave way to the usual lineup of Eagles, who were not affiliated with a major league club that season. Attendance for the game was 53,378, which shattered the previous Texas League record of 16,484. Here is a well-researched article about the game by Frank Jackson of The Hardball Times.
I had heard that the park was also used for an occasional exhibition game prior to the opening of Turnpike (later Arlington) Stadium about 20 miles to the west, and the eventual arrival of the Texas Rangers, but I have no corroboration of this.
I was not able to get into the Cotton Bowl on my 2005 visit.