Åberg represented team Stockholm at the Rikspucken, a national tournament in Sweden. He made his Elitserien debut during the 2010–11 season, as an extra forward in the second game of the season against Modo Hockey, but did not get any time on the ice. Åberg extended his contract with Djurgården to the 2013–14 season in May 2011. Åberg played a total of five games, scoring two points in the 2011 edition of the pre-season tournament European Trophy. Åberg received his first time on the ice in an Elitserien game and also scored his first goal in Elitserien in the league premier against HV71 on 15 September 2011. He scored the game-winning goal in the shootouts against Färjestad BK on 29 September 2011. After a start to the 2011–12 Elitserien season that included getting five goals and three assists in the first fourteen games, Åberg was announced to be the first of four candidates for that season's Elitserien Rookie of the Year. In the 2013–14 season, Åberg transferred to play for fellow SHL club, Färjestad BK.
NHL
Nashville Predators
After signing an entry-level contract with the Predators, he began his first North American season in 2014–15 season playing for the Milwaukee Admirals of the AHL. In the 2016–17 NHL season Åberg appeared in 15 regular season games posting 1 goal and 1 assist with the Predators. In the AHL Åberg was effective potting 31 goals in just 56 games, and he added 21 assists to bring his point total to 52 with the Admirals. After Ryan Johansen went down in Game 4 of the WCF he was announced to be out for the remainder of the playoffs, and with Predators captain Mike Fisher being sidelined as well, the Predators called upon Sissons and Åberg to help carry the offensive load. In game 5 Åberg scored the game-winning goal in Anaheim in a 3–1 win to give the Predators a 3–2 series lead. In Game 6 at Bridgestone Arena, Åberg assisted 2 of Sissons' 3 goals to help the Predators advance to their first ever Stanley Cup Finals. On 18 July 2017, the Predators re-signed Aberg to a two-year, $1.3 million contract worth $650,000 annually. During his time with the Predators', he was given the nickname "The Pope".
In the 2017–18 season, at the NHL trade deadline on 25 February 2018, Åberg was dealt by the Predators to the Edmonton Oilers in exchange for Mark Letestu. He made his Oilers debut on 27 February against the San Jose Sharks. He scored his first point as an Oiler in his debut with an assist on a goal from Jesse Puljujärvi, in a 5–2 loss to the Sharks. In remaining with the Oilers to play out the season, Åberg contributed with 2 goals and 8 points in 16 games.
After attending the Oilers 2018 training camp, Åberg was unable to retain his roster spot approaching the 2018–19 season. On 1 October 2018, he was claimed off waivers by the Anaheim Ducks. He made his debut with the Ducks in an opening night 5–2 victory over the San Jose Sharks on 3 October 2018. After a brief stint with AHL affiliate, the San Diego Gulls, Åberg returned to the Ducks scoring two goals in consecutive games on 28 October and 30 October against the Sharks and Philadelphia Flyers. Through 37 games with the Ducks, Åberg had already eclipsed career highs offensively, placing second on the team with 11 goals and recording 19 points before he was relegated as a healthy scratch through a franchise record losing streak.
On 16 January 2019, Åberg's tenure with the Ducks ended as he was dealt to the Minnesota Wild in exchange for Justin Kloos. Åberg was unable to replicate his offensive pace with the Wild, remaining on the roster through the conclusion of the regular season to post 1 goal and 6 points in 22 games. On 25 June 2019, Åberg was not tendered a qualifying offer by the Wild, releasing him to free agency.
On 24 July 2019, Åberg signed as a free agent to a one-year, $700,000 deal with the Toronto Maple Leafs. In the 2019–20 season, Åberg was primarily assigned to AHL affiliate, the Toronto Marlies. He was productive with the Marlies, registering 20 goals and 44 points in 55 games, while making just 5 appearances with the Maple Leafs adding 1 assist. With the remainder of the AHL season cancelled due to the COVID-19 pandemic, Åberg as an impending free agent returned to his native Sweden.
KHL
Left off the Maple Leafs return to play training camp roster, Åberg effectively left the NHL by signing a one-year contract with Russian club, Traktor Chelyabinsk of the KHL on 13 July 2020.
