Costly drove the ball home after 31 minutes when a rebound from a long ball dropped into his path.
Honduras, appearing at the finals for the third time, had last scored in the tournament when they drew 1-1 with Northern Ireland on June 21, 1982.
They had since gone five games at the World Cup without scoring, including a goalless campaign in 2010.
Their lead was short-lived as Ecuador equalised three minutes later through Enner Valencia.
(Writing by Keith Weir, Editing by Ed Osmond)
- Sports & Recreation
- World Cup finals
- Carlo Costly