Kinsler, 36, is batting .239 this season, with 13 home runs and 32 RBIs. But with second baseman Dustin Pedroia shelved indefinitely, the Red Sox had been looking for more depth at the position.
Red Sox president of baseball operations Dave Dombrowski told reporters after Monday's 2-1 win over the Philadelphia Phillies that he expects Kinsler to be the regular second baseman, moving Brock Holt and Eduardo Nunez back into utility roles once third baseman Rafael Devers comes off the disabled list.
"Ian's a really good defensive player," Dombrowski said. "Basically, it's what we hoped, early this season, that Dustin would be."
Kinsler, who was a finalist for the Gold Glove Award in 2017, has never started at another defensive position in his 13-year major league career.
"He brings that intensity, that fire every single day. He's a gamer. He'll make any team better. I'm very happy we got him."
Angels general manager Billy Eppler said he spoke with Kinsler and thanked him for his time with the team.
"That was difficult, but we're excited about the players we're getting back and the potential for an impact in the future," Eppler said on a conference call late Monday night. "There's a reason people gravitate to this guy. I know I did, and I was only around him less than a calendar year. We have to weigh that impact and the duration of that impact, for the guys returning in this deal."
The Angels are getting two Triple-A relievers, right-hander Ty Buttrey and left-hander Williams Jerez, in return.
A 25-year-old right-hander, Buttrey has a 2.25 ERA in 32 relief appearances at Triple-A Pawtucket this season.
"He looks pretty darn close to major-league-ready," Eppler said. "Looks like a guy that's got a lot of upside."
Jerez, 26, is 2-1 with five saves and a 3.63 ERA in 34 relief appearances for the PawSox. The left-hander averages 95 mph with his fastball and hits 100.
"We're hoping those guys take that next step and we watch them in Los Angeles," Eppler said. "The Angels will always remain opportunistic. That's what at least we hope this trade speaks."
The Angels are also sending the Red Sox $1.8 million of the remaining $3.6 million on Kinsler's deal, according to multiple reports.
Dombrowski is familiar with Kinsler. In November 2013, when he was general manager of the Tigers, Dombrowski traded for Kinsler, bringing him over from the Texas Rangers in a straight-up trade for Prince Fielder.
Kinsler played for the Tigers for four years. He was named to his fourth All-Star team in 2014, and in 2016 he won a Gold Glove.
Kinsler, who spent his first eight seasons with the Rangers, is a career .272 hitter. In December 2017, he was traded to the Angels.
He is the third former Tiger currently on the roster acquired by the Red Sox under Dombrowski, joining Price and slugger J.D. Martinez.
Dombrowski said the Red Sox could be done dealing, adding that he likes his bullpen, which has been speculated to be an area the team would address.
The Associated Press contributed to this report.