The trade was completed with a little time to spare before Wednesday's 4 p.m. ET non-waiver Trade Deadline.
Trailing the Rays by a half-game in the American League East, Cherington hopes Peavy can aid the final push for Boston's attempt at its first postseason berth since 2009.
"We're really excited to bring Jake here," said Cherington. "He's obviously a proven Major League starter. He's had a ton of success in his career. And I think if there's one thing we wanted to do -- if we could pull it off -- is to add a starting pitcher. As we looked at the next two months, we're in position to compete for a playoff spot and we just felt like adding a starting pitcher was probably the most important thing we could do to protect our chances to do that."
Boston also landed Detroit right-handed reliever Brayan Villarreal, who will report to Triple-A Pawtucket. Tigers outfielder Avisail Garcia, whom the Red Sox acquired in the deal, was then sent to the White Sox.
The Red Sox also sent Minor League right-handers J.B. Wendelken and Francelis Montas and Minor League infielder Cleuluis Rondon to the White Sox.
Peavy gives the Red Sox another proven arm in the rotation at a time when Clay Buchholz remains out with a right bursa sac strain.
Buchholz last pitched for the Red Sox on June 8, and he is likely three to four weeks from being activated.
"There's no new update on Clay since the last time we talked. He's continued to progress and is doing well," Cherington said. "As I said before, the situation with Clay, we've been aware of for a while, certainly it's been a factor to some degree in our discussions with other teams, but we still expect Clay to come back and be in our rotation.
"We just felt, again, that the most important thing we could do to protect the team, to give the team every chance to be strong over the last couple of months was to bolster the rotation. Assuming Clay does come back, we'll be even better at that point."
The 32-year-old Peavy is 8-4 with a 4.28 ERA in 13 starts. He has made two starts since returning from the disabled list. Peavy had been sidelined with broken ribs.
Iglesias has been playing mainly third base for the Red Sox, but he's a superb defender at shortstop.
The Tigers could soon see their starting shortstop Jhonny Peralta get suspended as part of the Biogenesis case that could impact several teams during the pennant race.
Will Middlebrooks, who was demoted to Triple-A on June 25, could be recalled by the Red Sox as early as Wednesday to fill the void left by Iglesias.
Though the Red Sox have been linked to the Phillies' Michael Young in some rumors, Cherington said he doesn't expect to trade for a third baseman before Wednesday's Deadline.
Iglesias got off to a hot start for the Red Sox at the plate this season, but has cooled off of late. He went 0-for-3 on Tuesday and is hitting .330 with one homer and 19 RBIs.
Boston signed Iglesias out of Cuba in 2009. He has always been highly touted for his glove, and this year has proved his bat also has some life in it.
Xander Bogaerts, the top position player in Boston's farm system, is a shortstop, perhaps making it easier for the Red Sox to put Iglesias in a trade.
"We certainly didn't go into this July looking or expecting to trade Jose, but we felt like given the strength at that position in the organization with Stephen Drew, obviously, our everyday shortstop in Boston right now and the guys we have coming in behind him in the Minor Leagues, it was a relative area of strength to deal from," said Cherington. "And if we could get the starting pitcher that we really wanted, ultimately, we thought it made sense to do the deal."
MLB Network contributor Jon Heyman of CBSSports.com was the first to report the Red Sox and White Sox had agreed to a deal. WEEI.com reported that the Tigers were also involved and could be getting Iglesias from Boston.
The Red Sox control Peavy's contractual rights for next season, when he will earn $14.5 million.
Peavy also has innings-based incentives that could kick in a $15 million option for the '15 season.
"The extra year of control is important. We went into the Deadline feeling like if we could make a move to improve the team and protect our chances to compete and give us every chance to get into October, we wanted to do that," said Cherington. "We wanted to do it in a way that's not all about this year, preferably, and we were able to find this deal that fit in that category."
Peavy joins a rotation that also includes Jon Lester, Felix Doubront, John Lackey, Ryan Dempster and, presumably, for the final few weeks of the season, Buchholz.
"Hopefully it puts our rotation in a good position this year," said Cherington. "It's an area of strength and it should be an area of strength as we move to 2014, too."
In 12 Major League seasons, Peavy is 128-97 with a 3.49 ERA. He won the National League Cy Young Award with the Padres in 2007, going 19-6 with a 2.54 ERA.
Peavy has been with the White Sox since 2009.
He should make his Red Sox debut within the next two to three days. Peavy would have started for the White Sox on Tuesday night, but he was scratched because of the trade possibility.
Cherington said Peavy would not pitch Wednesday, as Lackey will make that start.
For weeks, Cherington had spoken with the White Sox about Peavy. But the clubs didn't have a direct match. The Red Sox had no interest in including Middlebrooks in that swap.
Once the Tigers were included, things sped up.
"We had talked to the Tigers a little bit," Cherington said. "We talked to every team over the course of the days leading up to the Deadline, so we talked to the Tigers a little before today, but it really became more of a three-team picture today. I can't remember what time today, but it was pretty much today."
Though Peavy might not be as dominant as he was in '07, he was an All-Star as recently as last season.
"He's an intense competitor," Cherington said. "Loves to pitch. He's got a good assortment of stuff, a fastball with a ton of life, along with a good slider and changeup. He throws strikes. He attacks hitters and he loves to pitch. I think he'll fit in nicely with the group we have and he's obviously been very successful in his career."