The decision to retain both was in line with the club's plan to cut costs during its youth movement. Both players were on the parent roster at the end of 2002.
Mulholland, 39, signed a one-year contract.
Financial terms were not disclosed. He made $3 million in 2002.
Burba, 36, signed a minor league contract with an invitation to spring training.
His financial terms also were not disclosed. He made $5 million in 2002.
Mulholland started last season with Los Angeles and was acquired by the Indians on July 28. Overall in 2002, he was 3-2 with a 5.70 ERA in 37 games.
In a 17-year career, Mulholland is 116-127 with a 4.34 in 547 appearances, including 314 starts.
Burba was 5-5 with a 5.20 ERA in 35 appearances last season, including 21 starts. He began the season with Texas, but was releases in late July and signed with Cleveland in August.
The 13-year veteran has a career record of 110-85 with a 4.52 ERA in 443 appearances, including 232 starts.
To make room for Mulholland on the 40-man roster, the Indians designated pitcher Jose Santiago for assignment.
Gal Gadot cast as Wonder Woman for 'Batman vs. Superman'
Beyonce flaunts bikini body, Blue Ivy in vacation pics