Columbus Blue Jackets

Following his rookie season with the Blue Jackets, Moore was assigned to start the 2010–11 season with the Blue Jackets' American Hockey League (AHL) affiliate, the Springfield Falcons. After 47 games with the Falcons, Moore was recalled by the Blue Jackets and made his NHL debut in a 4–3 victory over the Edmonton Oilers on February 5, 2011.^[9] His first NHL goal was scored on October 25, 2011 against Ty Conklin of the Detroit Red Wings.^[10]

New York Rangers

During the lockout-shortened 2012–13 season, Moore was included in the NHL trade deadline day deal made by the Blue Jackets that sent Derick Brassard, Derek Dorsett, himself and a sixth-round draft pick to the New York Rangers in exchange for Marián Gáborík and prospects Blake Parlett and Steven Delisle on April 3, 2013.^[11] In his first game as a Ranger, only three hours after the trade, Moore scored his first goal of the season in a 6–1 victory over the Pittsburgh Penguins.^[12]

Arizona Coyotes

On March 1, 2015, Moore, along with Anthony Duclair, a 2015 second-round pick and a 2016 first-round pick, were traded to the Arizona Coyotes in exchange for defensemen Keith Yandle, Chris Summers and a 2015 fourth-round pick.^[13] In his short time with the club, Moore recorded 1 goal and 5 assists in 19 games. On June 29, Moore was not tendered an offer as a restricted free agent by the Coyotes, making him an unrestricted free agent.^[14]

New Jersey Devils

On July 1, 2015, Moore signed as a free agent to a three-year, $5 million contract with the New Jersey Devils.^[15]

During a game against the Washington Capitals in December 2016, Moore was awkwardly hit into the boards by Capitals forward Tom Wilson and stretchered off the ice.^[16] He missed 17 games to recover from a concussion before returning to the lineup on February 12, 2017, against the San Jose Sharks.^[17] When reflecting on his recovery, Moore stated he worked with Devils skills coach Pertti Hasanen and "really kind of challenged myself when I was injured."^[18]

Boston Bruins

On July 1, 2018, Moore signed as a free agent to a five-year, $13.75 million contract with the Boston Bruins.^[19] Following the signing, Moore stated "it was a slam dunk and something I wanted to happen...[Free agency] is a bizarre thing, but when [Bruins General Manager] Don [Sweeney] called it felt right and we just wanted to make it work, and I'm really grateful that we did."^[20]

Prior to rejoining the Bruins following an injury, Moore was re-assigned to the Providence Bruins on a conditioning stint on December 1, 2019.^[21]

Following the Bruins elimination from the 2021 Stanley Cup playoffs, Moore underwent hip arthroscopy and labral repair surgery.^[22]

On October 9, 2021, Moore was placed on waivers, which he cleared.^[23] However, on October 11, 2021, he was announced as part of the Bruins' opening night roster.

Anaheim Ducks

On March 19, 2022 Moore was involved in a trade that sent him to the Anaheim Ducks along with Urho Vaakanainen in exchange for Hampus Lindholm.^[24] The Ducks then attempted to trade both Moore and the remainder of Ryan Kesler's contract to the Vegas Golden Knights in exchange for Evgenii Dadonov and a conditional second-round pick at the NHL trade deadline on March 21, 2022. However, the deal went into dispute due to a no-trade clause in Dadonov's contract.^[25] On March 23, the NHL officially cancelled the trade, therefore Moore remained in the Ducks' organization.^[26]