Mesa was original signed as an amateur free agent by the Toronto Blue Jays in 1981, originally as an outfielder. He was traded to the Baltimore Orioles, however, during the 1987 season for starting pitcher Mike Flanagan. Mesa began his major-league career as a starting pitcher with the Orioles who hoped that Mesa's superior fastball would earn him success in the starting role. After 48 starts of below-average results during four seasons, the Orioles traded Mesa to the Cleveland Indians in 1992. Cleveland continued to use Mesa as a starter through the end of 1993; that year, Mesa pitched a career-high 208-2/3 innings, although Mesa's earned-run average was worse than the league average ERA for the fifth consecutive season.

Mesa became a relief pitcher for the Indians in 1994, and, for the first season of his career, he posted an ERA better than the league average. In the role of closer during the 1995 season, Mesa pitched superbly; in 64 innings pitched over 62 appearances, Mesa had a 1.12 ERA, saved 46 games, and won 3. Thirty-eight of Mesa's 46 saves were recorded in consecutive appearances in save situations; this was a major-league record at the time. Mesa's performance in 1995 was instrumental to the Indians' 100-44 regular-season record and their first World Series appearance since 1954. In 1997, Mesa's 2.40 ERA, 16 saves, and 4 wins helped the Indians to their second World Series appearance in three seasons. However his inability to hold a one run lead in Game 7 of the 1997 World Series was a black mark on his career, and Cleveland traded him mid-way through the following season.