Burris was born in Idabel, Oklahoma. The son of sharecroppers, Burris spent up to fifteen hours a day working the fields with his family through high school. He earned a Bachelor of Arts degree in Recreational Leadership at Southwestern Oklahoma State University, in addition to playing basketball and baseball. In 1972, he received All-America honors and set a National Association of Intercollegiate Athletics record with 150 strikeouts, and pitched the Bulldogs to a fifth-place finish at the NAIA National Tournament.
Burris was drafted by the Chicago Cubs in the 17th round of the 1972 Major League Baseball draft. After just one season in the Cubs' farm system, in which he went 7–5 with a 3.51 earned run average for the Texas League's Midland Cubs, Burris jumped to the majors to start the 1973 season. He made his major league debut in the third game of the season against the Montreal Expos, and pitched 3.1 innings of scoreless ball. In his next appearance, he made his only start of the season. He pitched five innings of shutout ball to defeat Jon Matlack and the New York Mets. He did not allow an earned run until his sixth appearance of the season, against the San Francisco Giants. Overall, Burris went 1–1 with a 2.92 ERA.
He returned to the bullpen in 1974. His first major league save was an impressive six out save, in which he struck out four of the six Pittsburgh Pirates batters he faced to secure the 1–0 victory. Two days later, he entered a game against the Philadelphia Phillies in the 11th inning with runners on first and second and one out. He struck out Terry Harmon, and got Bob Boone to hit a weak pop up to first base to escape the inning, then pitched two more scoreless innings to earn his second win of the season, and lower his ERA to 0.93.

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