After attending Oklahoma State University, Stanka spent most of his career in the minor leagues, making his Major League debut with the White Sox in 1959 at age 28. His big-league career consisted of only two appearances that year. In his first MLB game, September 2 against the Detroit Tigers, Stanka entered the game in relief of starting pitcher Barry Latman in the fifth inning at Comiskey Park with Chicago trailing 3–0. He retired the Tigers without further damage. Then, in their half of the fifth, the White Sox exploded for 11 runs, with Stanka contributing to the rally with a single in his first big-league at bat. He went on to pitch 3 ¹⁄₃ innings of one-hit, one-run relief and was credited with the win in an 11–4 ChiSox triumph. Stanka pitched in only one more contest that month, a two-inning relief stint against the Cleveland Indians, September 5. The White Sox and Indians were then embroiled in a pennant race that ultimately delivered Chicago its first American League title since 1959. Stanka did not appear in the 1959 World Series.

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