Veselic made his MLB debut on September 18, 1980, in relief of Pete Redfern in a 5–0 loss to the Milwaukee Brewers. After spending 1981 with the Triple-AToledo Mud Hens, Veselic again joined the Twins in September and worked in five games. On September 13 at Metropolitan Stadium, he relieved starter Don Cooper in the sixth inning with the Twins trailing the Chicago White Sox, 4–2. After allowing one inherited runner to score on a sacrifice fly, he gave up only one more run the rest of the way, and the Twins rallied to win the game, 7–6, for Veselic's only Major League victory.[1] (Ironically, Cooper later became the White Sox' longtime pitching coach.)
Veselic returned to the minors in 1982 and finished his career after the 1983 season. He died from cancer[2] at the age of 40 in Fontana, California.