Strome began his professional playing career with the Toronto Marlboros of the Greater Toronto Hockey League (GTHL) during the 2008–09 season. At the Ontario Hockey League’s (OHL) 2009 Priority Selection, he was selected eighth overall by the Barrie Colts. During the 2009–10 season, he was traded to the Niagara IceDogs. The IceDogs qualified for the playoffs but were eliminated in the first round, while the Colts would go on to the OHL Championship Finals but were eliminated in four games. During the following season, Strome compiled 106 points in just 65 games. He was named “OHL Player of the Month” in January 2011 and selected to the Second All-Star Team at years end.

After his second season with the IceDogs, Strome became eligible for the National Hockey League (NHL) Entry Draft. He was selected in the first round, fifth overall, in the 2011 draft by the New York Islanders. He was also selected by the Kontinental Hockey League’s Amur Khabarovsk in their 2011 draft. After being drafted, Strome spent another season with the IceDogs before joining the Islanders’ American Hockey League (AHL) minor league affiliate, the Bridgeport Sound Tigers in the 2012–13 season. Halfway through the following season, on December 11, 2013, he was called up to the Islanders after the team had put Pierre-Marc Bouchard on waivers to make room on their roster. Before his call-up, he was leading the AHL in points through the first 23 games of the season. Strome scored his first NHL goal on January 6, 2014 against Kari Lehtonen of the Dallas Stars.

On June 22, 2017, Strome was traded by the Islanders to the Edmonton Oilers in exchange for forward Jordan Eberle. In his first season with the Oilers in 2017–18, Strome appeared in all 82 games posting 13 goals and 21 assists for 34 points. As a restricted free agent in the off-season, Strome agreed to two-year, $6.2 million extension with the Oilers on July 6, 2018.

Strome has two brothers who also play hockey, Dylan and Matthew. Dylan currently plays in the National Hockey League with the Chicago Blackhawks organization. The youngest brother Matthew was drafted by the Philadelphia Flyers in 2017. The Strome family had been dubbed as the future Sutters and Staals, families where multiple siblings have gone on to successful ice hockey careers,. The Strome brothers grew up in the Lorne Park area of Mississauga.

Strome was traded by the Oilers to the New York Rangers in exchange for forward Ryan Spooner on November 16, 2018.

On November 6, 2020, Strome signed a two-year, $9 million contract with the Rangers.