Left-handed hitters are batting .173 against Thornton this season compared to .320 against right-handers, though there's been less of a differential in the past over his 10-year career, with lefties batting .229 and righties hitting .238.
"It is down a little bit from the past, but he's still getting lefties out," general manager Ben Cherington said. "It used to be elite stuff. It may just be really good stuff now. We're confident he has plenty of stuff to be effective in the role we need him to be in and be an important part of the bullpen. He just needs to come in and be who he is."
Boston also acquired cash considerations in the deal, and it transferred Miller to the 60-day disabled list to make room for Thornton, a 2010 American League All-Star, on the 40-man roster.
"We are what we are, and we've had some attrition in that area. When Andrew went down, it probably increased our level of urgency," Cherington said. "We found [a fit] we think makes sense. There was some motivation to do it prior to the break, because after the break, we get back into AL East play. Stabilizing and fortifying the 'pen made sense."
Thornton, 36, is 0-3 with a 3.86 ERA in 40 appearances this season with Chicago. The veteran has recorded 18 holds this season, which is tied for fourth most among all AL relievers.
Farrell said Thornton will be used mostly in sixth- and seventh-inning situations -- similar to his role with the White Sox, who were rained out Friday in Philadelphia. Thornton is expected to be available in a Red Sox uniform in time for Saturday night's game against the A's in Oakland.
Jacobs was a 10th-round selection by the Red Sox in the 2009 First-Year Player Draft. He started the year with Class A Salem and was promoted to Double-A Portland in June.
The 23-year-old, who was ranked the organization's No. 11 prospect, combined to hit .246 in 83 games between the two clubs, with 24 doubles, one triple, 11 home runs, 44 RBIs, 33 walks and 90 strikeouts.
Thornton is owed close to $3 million for the remainder of the season, with the Red Sox holding a $6 million option for next season with a $1 million buyout. The cash considerations could be used to pay a portion of his remaining salary.
"We're in good position, and we have an obligation to do what we can," Cherington said. "We have some time before the deadline, and we're going to continue to look around at things to get better if something makes sense."