Chicago sent Triple-A slugger Dan Vogelbach and Double-A pitcher Paul Blackburn to Seattle, which shipped Triple-A pitcher Jordan Pries to the Cubs as part of the trade.
Montgomery, 27, immediately fills a need for the Cubs, who have been lacking a lefty reliever. He also has 18 starts in his two-year career, including two this season.
"I was a little sad at first, understanding that I'm not going to be here with these guys in this city. But at the same time, I'm excited because I'm going to a good team and I'm ready to go contribute any way I can and help them win a World Series,'' Montgomery said.
"It kind of hit me by surprise. When I heard the news, I wasn't shocked, but I definitely didn't see it coming."
Montgomery said the Cubs told him he would begin in the bullpen.
"Montgomery is somebody that has really good stuff and has been performing well this year. We think he's coming into his own a little bit," Cubs president of baseball operations Theo Epstein said. "He's someone who can help us this year and long into the future.
"We paid a price to do that. We traded a couple of really good prospects for him."
Lefties are hitting .164 off Montgomery in 2016, and he has a 2.34 ERA in 32 games, including 30 in relief. Epstein likened him to Andrew Miller when the now-dominant southpaw was just emerging for the Red Sox.
"Sometimes with these bullpen pieces, it's important to get them on the way there, maybe hasn't fully arrived yet," Epstein said. "I'm not saying he's Andrew Miller, very few are, but we traded for Andrew Miller in Boston when I was there in November 2010, hoping he could put it together in the bullpen someday. And he did. That's how some guys are. If you wait until they are fully established the price tag is virtually impossible to acquire."
Maddon described Montgomery as a "big left-hander with a very good arm."
Epstein said Montgomery "was always on our radar," and that while the lefty is not a household name, the club believes "he has a chance to take off and maybe be the type of guy that, a year from now, you couldn't get in a deal of this size."
"He's a guy with a plus fastball, been working in the mid-90s this year. Even in his last two appearances as a starter, he's held the velocity the entire time out there," Epstein said. "He's got a plus curveball, a changeup that we like, and he's been working on a cutter. He's got some swing-and-miss secondary weapons along with the power, and he's done a good job throwing strikes this year."
Vogelbach, whom the Cubs selected in the second round of the 2011 amateur draft, was being blocked at the big league level by Cubs first baseman Anthony Rizzo. At 6 feet, 250 pounds, Vogelbach's body type doesn't allow for another position besides designated hitter.
"Dan has done a great job. Sad to see Dan go. ... But also happy for him to get an opportunity, going to an American League club," Epstein said. "This is going to be a chance for him to get at-bats at the big league level. I'm sure he'll be up here pretty soon and he's going to be around for a really long time. He can hit and he's got great energy."
Dipoto said there was "no specific ETA" for Vogelbach's big league arrival but lauded the long-term value he brings to the team.
"He's someone who can help us this year and long into the future. We paid a price to do that. We traded a couple of really good prospects for him." Cubs president Theo Epstein on LHP Mike Montgomery
"Whether he fits on our roster today is a different matter," Dipoto said. "But what he brings us long-term is too good to walk away from."
Vogelbach, 23, was hitting .318 with 16 homers and 64 RBIs in 89 games for Triple-A Iowa.
"He rakes," Dipoto said. "He rakes everywhere he's ever been. He's an elite strike-zone controller with well above-average power. He has absolutely tormented right-hand pitching, especially this year."
Dipoto said the left-handed offense Vogelbach brings "was really too appealing to pass up."
Pries had a 7-3 record with a 4.93 ERA between Seattle's Double-A Jackson and Triple-A Tacoma this season. According to Epstein, the 26-year-old right-hander can fit as either a starter or a reliever, and the club plans to move him into the rotation at Triple-A.
"He's got a good slider. He can protect our rotation," Epstein said.
Blackburn, 22, had a 3.17 ERA and was 6-4 in 18 starts with Double-A Tennessee.
Dipoto said Blackburn "gives us another piece to build forward with, and we start to build up that next phase of our roster."
Information from The Associated Press was used in this report.