Texas also acquired minor league right-hander Jason Bahr from San Francisco, which received a player to be named later or cash considerations.
Jackson, 31, batted .242 with 13 RBIs in 59 games this season for the Giants. Gearrin, 32, posted a 4.20 ERA in 35 relief appearances with San Francisco.
Bahr, 23, has been assigned to the Class A Carolina League. He has an 8-4 record with a 2.55 ERA in 16 Class A starts this season.
The Giants parted with Jackson and Gearrin to create space on their 40-man roster for right-hander Ray Black and outfielder Steven Duggar, who had their contracts purchased and will be available for Sunday's game against the St. Louis Cardinals.
The Associated Press contributed to this report.