Watson was picked by the St. Louis Cardinals in the first round of the 1991 amateur entry draft. He spent the next two years in the Cardinals' minor league system, making his Major League debut on July 8, 1993.
In 1995, he was traded, along with Doug Creek and Rich DeLucia, to the San Francisco Giants for Chris Wimmer and Royce Clayton. At the end of the 1996 season, he and Fausto Macey were traded by the Giants to the Anaheim Angels for J. T. Snow. He pitched with the Angels in 1997-1998, obtaining a career-high twelve wins as a starting pitcher in 1997. On June 14, 1997, Watson gave up the first-ever grand slam in interleague play to Rich Aurilia of the San Francisco Giants
On July 3, 1999 Watson signed with the New York Yankees. Watson was used as a reliever during his time with the Yankees, and had his best tenure with them, going 4–0 with a 2.10 ERA to finish the 1999 season. His play earned him a spot on the postseason roster, where he pitched one inning in the 1999 American League Championship Series against the Red Sox. Watson gave up two hits, two walks, and one strikeout as the Yankees defeated Boston. Watson did not pitch in the 1999 World Series, but earned a ring regardless. On November 5, 1999, he was granted free agency, but he re-signed with the Yankees on December 7.
In 2000, Watson's ERA ballooned to 10.23 and he appeared in only 17 games. During which he pitched 22 innings and did not record a win or loss, while finishing only 9 games for the team. Despite his dismal performance during the season, he was placed on the Yankees postseason roster, but did not appear in any games. He earned his second World Series ring when the Yankees defeated the Mets in five games. Watson later had surgery to repair his shoulder, which caused him to miss the entire 2001 season. Watson returned to the Yankees during spring training in 2002, but made only one start. He retired after spring training.
Watson is currently working as a personal pitching coach for prospective athletes in Queens and Long Island. He works primarily out of The Cage located on Metropolitan Avenue in Ridgewood. Watson also runs baseball clinics for all ages in Oceanside, NY at South Shore Sports Complex and at East Coast Sports Academy.
On December 20, 2007, Watson was named in Jason Grimsley's affidavit as having used performance-enhancing drugs. Watson and Grimsley were teammates on the 1999-2000 New York Yankees. In a statement released by his agent, Watson denied these accusations by stating:
I at no time over my professional baseball career used steroids or any performance-enhancing drugs. Not then, not now, not ever.
In their baseball memoir The Yankee Years, Joe Torre and Tom Verducci recount an incident where Watson was horsing around in the Yankee clubhouse and threw a bagel towards a clubhouse attendant, just as Yankee owner George Steinbrenner walked in. When the bagel hit Steinbrenner, he demanded to know who threw it. When Watson confessed, Steinbrenner remarked "I figured it was you, Watson. That's why it didn't hurt."