Koskinen was selected by the New York Islanders in the first round (No. 31) of the 2009 NHL Draft after playing for Espoo in the Finnish Elite League. He signed a three-year, entry-level contract with the Islanders on July 13 and was assigned to Bridgeport of the American Hockey League to start the 2009-10 season but tore a labrum in his hip during his second game and missed most of the season. He returned March 19 and was reassigned to Utah of the ECHL. Undefeated in six starts, Koskinen helped Utah reach the conference semifinals and returned to Bridgeport for the first round of the Calder Cup Playoffs.
Koskinen made his NHL debut for the Islanders, a 5-3 loss to the Toronto Maple Leafs, on Feb. 8, 2011. He won his first NHL game two days later by defeating the Montreal Canadiens 4-3 in a shootout and returned to Finland by signing with KalPa on Nov. 12.