Burke was executive vice president and general manager of the Royals from the middle of the 1974 season through October 1981. He then served as club president until his death on May 12, 1992. During his tenure, Burke was general manager of the Royals' first American League championship team, the 1980 edition, then was president of the 1985 Royals, who won the franchise's first World Series title. In addition to those two pennant-winners, the Royals won American League West Division championships in 1976, 1977, 1978, 1981 (second half of a split season) and 1984. In 1976, he was named Major League Executive of the Year by The Sporting News after his first division title.
Before coming to Kansas City, Burke had been a member of the front office of the Texas Rangers and its predecessor, the second Washington Senators franchise. He had begun his baseball career in 1948 with the Louisville Colonels of the Triple-A American Association, where he worked under general manager Ed Doherty. After rising to the post of GM of the Colonels in 1960, Burke joined the expansion Senators in their debut 1961 season as business manager, again working for Doherty, the team's first general manager. He later was named the Senators' vice president and treasurer, and was retained when Bob Short purchased the Senators in 1968. He accompanied the franchise to Dallas-Fort Worth when it relocated after the 1971 season and became the Rangers' general manager in their first season in North Texas.
In June 1974, Burke became the second general manager in the Royals' six-year history. One of his first major moves was the hiring of Whitey Herzog as manager during the middle of the 1975 season on July 25. Herzog would be elected to the Baseball Hall of Fame as a manager in 2010, but he had failed dismally as the Rangers' pilot, working under Burke, during 1973. In Kansas City, he would turn the Royals into consistent contenders in the AL West. Burke also appointed Jim Frey and Dick Howser as managers after Herzog's exit, and each man would lead Kansas City to an American League pennant (and, in Howser's case, the 1985 World Series title as well).