Glavin first played hurling and Gaelic football at club level with Midleton.[2] After progressing through the juvenile and underage ranks, his first success at adult level came in 1978 when Midleton won the Cork IHC title. He won the first of three Cork SHC titles and the first of two Munster Club SHC titles in 1983. Glavin had one of his most successful club seasons in 1984 when he captained the Midleton junior team to the Cork JAHC before winning a Cork IFC title. He was also part of the Imokilly divisional team that won the Cork SFC title in 1984.[3] Glavin achieved the ultimate club success when Midleton beat Gort in the 1988 All-Ireland senior club final.[4] He later transferred to Ballincollig and won a second Cork IFC with his adopted club in 1994.
Glavin first appeared on the inter-county scene with Cork as a member of the junior team. He won a Munster JFC title in 1984, however, he later lost his place on the starting fifteen and was listed amongst the substitutes when Cork beat Warwickshire in the 1984 All-Ireland junior final.[5] Glavin earned a call-up to the Cork senior team following this success, and he made a number of appearances during the opening stages of the 1984–85 National League. He lined out with the junior team in 1985.[6]