Cole, a 29-year-old defenseman, was traded to the Senators as part of a three-team deal that sent center Derick Brassard to the Pittsburgh Penguins on Friday.
"Columbus is a tremendous organization," Cole told TSN. "They're a great team that's had a ton of success. I'm excited to get there and get going."
Cole had 13 points (three goals, 10 assists) in 47 games for Pittsburgh this season. He has 90 points (18 goals, 72 assists) in 385 games in eight seasons with the St. Louis Blues and Penguins, and 14 points (one goal, 13 assists) in 56 playoff games. He was part of Pittsburgh's Stanley Cup championship teams the past two seasons.
"It's certainly tough," Cole said. "Not only did I make a ton of great friends and win championships, it's obviously a great hockey team with a very good chance to three-peat. I would have loved to finish that out, but that said, Columbus is a great organization that I've admired from afar."
Cole, chosen by the St. Louis Blues with the No. 18 pick in the 2008 NHL Draft, is in the final season of a three-year contract with an average annual value of $2.1 million and can become an unrestricted free agent July 1.
"Ian Cole is a hard-working, competitive defenseman who we know very well having drafted him in St. Louis," Columbus general manager Jarmo Kekalainen told the Blue Jackets website. "He is a very good defender and a two-time Stanley Cup champion who can bring a lot of intangibles to our group."
Moutrey, 22, was a fourth-round pick (No. 105) by the Blue Jackets in the 2013 NHL Draft. He had 34 points (17 goals, 17 assists) in 136 games with Cleveland of the American Hockey League.
The Blue Jackets (31-26-5, 67 points) are two points ahead of the New York Islanders for the second wild card into the Stanley Cup Playoffs from the Eastern Conference entering their game against the Washington Capitals at Nationwide Arena on Monday (7 p.m. ET; SN, FS-O, NBCSWA, NHL.TV).
The Senators (21-30-10, 52 points) have lost four in a row and trail the Blue Jackets by 15 points.