Montreal Canadiens (2010–2016)
On 17 June 2010, Eller was traded by the Blues to the Montreal Canadiens, along with Ian Schultz, in exchange for goaltender Jaroslav Halák. He participated in the Canadiens pre-season games where he played both a center and winger role. As the 2010–11 season approached, Eller was assigned to the Canadiens top line alongside Tomáš Plekanec and Michael Cammalleri. Once the season began, Eller was assigned to a winger position but was held pointless through the first five games of the season. His drought continued as he remained goalless through his first 20 games before scoring his first as a Canadien against the Los Angeles Kings on 24 November. Eller was a healthy scratch for three games in mid-January before rejoining the lineup on 18 January to play alongside Plekanec and Andrei Kostitsyn. Upon re-joining the lineup, Eller surpassed his previous career-high of ice time by playing 16:53 minutes in a game against the Buffalo Sabres. In late February, Eller found chemistry alongside Travis Moen and Kostitsyn and the three carried a five-game, seven-point streak by 4 March. As the Canadiens faced off against the Boston Bruins in the first round of the 2011 Stanley Cup playoffs, Eller was given the responsibility of shutting down the Bruins’ top line. He finished third on the team with 18 hits during the postseason while also playing through a dislocated shoulder.
During the off-season, Eller underwent shoulder surgery but rejoined the team for their 2011–12 season home opener. He joined the team for their home opener on 13 October 2011 against the Calgary Flames where he played 10:57 minutes of ice time. He was reunited with his former linemates Andrei Kostitsyn and Travis Moen and the three combined for 10 points in the first five games. On 23 November, Eller tallied his first shorthanded NHL goal in a win over the Carolina Hurricanes. In December, he sat as a healthy scratch for one game before returning to the lineup centering Erik Cole and Max Pacioretty. He also received a $2,500 fine for boarding Los Angeles Kings defenceman Drew Doughty during a game. On 4 January 2012, Eller scored four goals and an assist in a 7–3 victory against the Winnipeg Jets, his first career hat-trick. He subsequently became the first Montreal Canadien to score four goals at home in a single game since 27 February 1993. He continued to improve from his rookie season with the Canadiens and surpassed his 2010–11 season total in only 43 games. He finished his second season with the Canadiens with career-highs in games played, goals, assists, and points. As such, he signed a two-year contract to remain with the Canadiens through the 2012–13 and 2013–14 seasons. Once the Canadiens 2011–12 season ended, Eller returned to his home country to conduct his off-season training.
During the 2012–13 2012–13 NHL lockout, Eller signed a temporary contract with JYP Jyväskylä in the Finnish elite hockey league Liiga. While with JYP, Eller played alongside Boston Bruins forward Rich Peverley and ex-NHLers Éric Perrin and Ramzi Abid. He recorded five goals and 10 assists through 15 games before returning to the Canadiens in January 2013. Upon re-joining the team, Eller bounced between different lines before centering Colby Armstrong and Moen. He also earned a spot on the penalty kill as a replacement for an ill Max Pacioretty where he tallied a goal and two assists while going 63% in the faceoff circle during a 6–1 win over the Sabres in early February. Later that month, Eller was paired alongside Alex Galchenyuk and Erik Cole and they immediately made an impact on the team. By April 2013, Eller had accumulated four goals and 19 points through 35 games. During the 2013 Stanley Cup playoffs, Eller was knocked unconscious following an open ice check by Ottawa Senators' defenceman Eric Gryba. He was stretchered off the ice and was later diagnosed with a concussion, along with dental and facial fractures.
Eller recovered from the injury during the off-season and returned to the Montreal Canadiens lineup to begin the 2013–14 season. During the off-season, Eller and his agent were in talks with the Canadiens organization on a contract extension but eventually decided to see his play during the 2013–14 season. Once the season began, Eller was named the Second Star of the Week for the week ending on 6 October. He received this honor after factoring in on five of Montreal's seven goals and becoming the third Canadiens player since 2002–03 to register three or more points in the team's season opener. Eller also started the 2013–14 season shooting over 42% in his first five games before seeing his conversion rate fall toward his career average of 11.3%. In mid-October, Eller was transitioned from winger to center and he centered the "Kid Line" between Alex Galchenyuk and Brendan Gallagher. While on this line, Eller tallied nine points in 13 games during the month of October and led the team in goals during the early stages of the season. Due to this, he surpassed his goal tally from the previous season in five less games by scoring his 9th of the season in a game against the Carolina Hurricanes on 31 December. Eller finished the regular season with 12 goals and 14 assists including two goals on the power play, one while shorthanded and scored three game-winning goals. He also led all Canadiens forwards with 130 hits and recorded 137 shots on goal, while averaging 15:57 of ice time per game.
As the Canadiens qualified for the 2014 Stanley Cup playoffs, Eller continued to produce and set career-highs in games played, goals, assists, and points. During the first round of the playoffs, Eller amassed at least a point per game and finished with six points in five playoff games against the Tampa Bay Lightning. During game one of the second round against the Boston Bruins, Eller scored the first goal of the series to help lead the Canadiens to a win. They eventually fell to the New York Rangers in game six of the Eastern Conference finals. On 24 July 2014, Eller signed a four-year, $13 million contract extension with the Canadiens.
Following the signing of his contract extension, Eller spent six weeks in his hometown of Copenhagen before returning to Montreal to resume his daily workouts at the Bell Sports Complex. Once the 2014–15 season began, Eller centered Jiří Sekáč and Brandon Prust through the first half of the season but when his production began to wane, his name was flown around to be traded. However, he remained with the team and played with Devante Smith-Pelly and Jacob de la Rose over the final six games of the season. While with his new linemates, Eller snapped a 14-game goalless streak on 7 March and combined for numerous shots on net and hits.
