Tatum's professional career began when he signed with the Angels in 1966 after he was selected in the second round of the secondary phase of the 1966 Major League Baseball Draft.

In his fourth minor league season, in 1969, he was converted from a starting pitcher to a relief role, and he was recalled by the Angels in May.

During the season's first two months, he appeared in 20 games, won two of three decisions, netted eight saves, and posted a 1.00 ERA in 27 innings pitched.

'"[7] Ten days after the 1970 season ended, the Angels traded Tatum to the Red Sox in a six-player swap that brought slugger Tony Conigliaro to Anaheim.

(Ironically, Conigliaro's career was curtailed by the after-effects of a beanball in a game between the Red Sox and Angels on August 18, 1967.)

In his first game, on April 8, 1971, he blew a 2–1 ninth-inning lead for Sonny Siebert and lost to the Cleveland Indians on a walk-off single to Gomer Hodge.

He was dealt along with Reggie Smith from the Red Sox to the St. Louis Cardinals for Bernie Carbo and Rick Wise on October 26, 1973.