The Red Sox made their fifth trade in the last month and will send right-handed pitcher Pat Light to the Twins.
Kimbrel currently is recovering from a knee injury that required surgery on July 11. Uehara, who assumed closing duties for Kimbrel, has been on the disabled list since July 20 with a right pectoral strain.
Abad was 1-4 with a 2.65 ERA in 39 appearances with Minnesota. It is the second-lowest ERA of his career, which began in 2010 in the Houston Astros.
This year, Abad has held hitters to a .220 average and left-handed batters are batting .163 (8-for-49) against him.
Abad is making $1.25 million this season and is under team control through 2017 via arbitration. He has a 3.58 ERA and a 1.29 WHIP in 297 career appearances with Houston, Washington, Oakland and Minnesota.
Light made two appearances for the Red Sox earlier this season and allowed eight runs in 2 2/3 innings. In 25 appearances with Triple-A Pawtucket, the 25-year-old is 1-1 with a 2.32 ERA and at times his velocity has reached 100 mph.