In 1941 he moved from Kirkintilloch Rob Roy to Glasgow Rangers, and within eighteen months he was playing for Scotland; in one of his first matches, they lost 8–0 to England and he was dropped, but he soon found his way back into the team. He went on to captain Rangers to six Scottish League championships, two league cups, and four Scottish cups and was chosen Scottish Player of the Year in 1955.

Rangers' success during this period owed much to its tight defence, known as the ‘Iron Curtain’, in which Young played at centre half or right back. Standing at 6 feet 2 inches, he dominated battles in the air, but also possessed nifty control on the ground for someone who weighed more than 15 stone, and was known for his clean tackling. He was never sent off during a match and was booked only once.

Young's nicknames included Gentle Giant, Rock of the North, and Corky, the last because he always carried around a champagne cork that reminded him of his first game against England, a 3–1 victory at Wembley in 1949.  Between 1943 and 1957 he played in fifty-three internationals, a total eclipsed by Denis Law only when international fixtures were becoming more frequent.

From 1959 until 1962 Young served as manager of Third Lanark, whom he took to third place in the first division.  In 1963 Young—along with Ted Drake, Tom Finney, Tommy Lawton, and Arthur Ellis, a former referee—was one of the original ‘pools panel’, gathered together to forecast the results after three successive coupons had been declared void owing to the weather. He also organized tournaments and stadium tours for youngsters from abroad and wrote a column for The People.

Young spent his final years in a nursing home in Slamannan, near Falkirk. He died on 10 January 1997 in the Royal Infirmary, Falkirk. There was a minute's silence before Rangers' game against Aberdeen at Ibrox two days later.