The right-hander broke the tie that he had shared with Octavio Dotel of 13 teams.
He signed a minor league contract with Oakland last month and was 1-0 with a 6.75 ERA in three games — two at Triple-A and one at Class A.
Jackson has gone 104-123 with a 4.60 ERA in 394 career games. He made his big league debut with the Los Angeles Dodgers in 2003 and also pitched for Tampa Bay, Detroit, the Chicago White Sox, Arizona, St. Louis, Washington, the Chicago Cubs, Atlanta, San Diego, Miami, Baltimore and Oakland.
He was 6-3 with a 3.33 ERA in 17 starts for the Athletics last season.