Engomitis was born on May 26, 1972 in Ammochostos. He started his career at Ethnikos Achna and transferred to Anorthosis in 1997. He was acquired by PAOK Thessaloniki after the 2000 season and played in Greece until 2006, when he returned to Ethnikos Achna.
In July 2009, Engomitis announced his retirement from football.
Engomitis was capped 45 times by the Cyprus National Football Team.