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.