Finley made his MLB debut with the Angels on May 29, 1986, during a 7–4 loss to the Detroit Tigers. In relief of starting pitcher Don Sutton, Finley allowed two runs on two hits in one inning pitched. He also recorded his first major-league strikeout, coming against Pat Sheridan. Finley made a total of 25 appearances, all in relief, with the 1986 Angels, pitching to a 3.30 ERA and a 3–1 record while striking out 37 batters in innings. In his only postseason appearances with the Angels, Finley played in three games of the 1986 American League Championship Series against the Boston Red Sox, allowing just one hit in two innings pitched.