Burke first came to prominence on the inter-county scene as a member of the Cork team that won a second consecutive Munster title in 1927 with a 5-3 to 3-4 victory over Clare. It was his first provincial winners' medal. The subsequent All-Ireland final saw Cork take on Burke's native county of Dublin once again. Cork fell behind by 2-3 to 0-1 at half-time; however, they fought back in the second-half. In a team made up of nine members of the Garda Síochána ‘the Dubs’ claimed the victory by 4-8 to 1-3.
In 1928 Cork faced Clare in the Munster final for the second year in-a-row. That game ended in a draw, however, in the replay Cork triumphed with Burke collecting his second consecutive Munster title. Cork later defeated Dublin in the All-Ireland semi-final before lining out against Galway in the championship decider. Galway got a bye into the final without picking up a hurley, however, the game turned into a rout. A score line of 6-12 to 1-0 gave Cork the victory and gave Burke an All-Ireland winners' medal.
In 1929 Cork retained their provincial dominance for a fourth consecutive year. A 4-6 to 2-3 defeat of Waterford gave Burke his third Munster title in three provincial campaigns. The subsequent All-Ireland final was a replay of the previous year’s game as Cork played Galway once again. Mick Ahern scored a goal for Cork after just 25 seconds to start another rout. Cork won the day by 4-9 to 1-3 giving Burke his second All-Ireland title. It was his last appearance with Cork.
- Corry, Eoghan (2005). The GAA Book of Lists. Hodder Headline Ireland. p. 344.
- The GAA Book of Lists p. 345