Born in Saskatoon, Saskatchewan, Leswick was noticed by the Rangers when he toiled for the senior Saskatoon Wesleys in his home town. He scored a goal in his only NHL game with the Blueshirts in 1936-37 and spent most of his rookie pro season with the AHL's New Haven Eagles. He then split the next seven seasons between the Pacific Coast Hockey League (PCHL) and the American Hockey Association (AHA). Leswick contributed to the Seattle Seahawks PCHL championship in 1938 then led all playoff scorers six years later when leading the New Westminster Royals to the title. In 1942 he was voted on to the AHA first all-star team after scoring 35 goals for the Fort Worth Rangers.