Signed out of Brigham Young by the Mariners, Nielsen recovered from 1985 arm surgery to debut with the 1986 Yankees. He led their weak staff with two shutouts. Shipped to the White Sox in January 1987, he was reacquired with Rich Dotson in November. Nielsen holds the distinction of winning a 17-0 decision over the Cleveland Indians on July 5, 1987. That equaled a White Sox club record for the widest shutout margin against an opponent. On June 9, 1988 he pitched the first nine-inning no-hitter in Columbus (International League) history, defeating Maine 3-0.