The right-handed pitcher appeared in Major League Baseball in six games and 262⁄3 innings, all in relief, for the 1980–1981 Minnesota Twins.

Veselic was native of Pittsburgh, Pennsylvania, but attended Alameda High School in the San Francisco Bay Area.

Veselic made his MLB debut on September 18, 1980, in relief of Pete Redfern in a 5–0 loss to the Milwaukee Brewers.

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.