Messi scored twice after Antoine Griezmann had opened the scoring on the hour mark, before Frenkie de Jong headed in a second goal three minutes later.
Messi dealt a double blow to put the result beyond any doubt by the 72nd. The first of the Argentine’s two goals was a magnificent effort as the 33-year-old picked up the ball in his own half, travelled towards the penalty area, played two one-twos with teammates and then jinked inside a defender to finish into the far corner.
It is Barcelona’s first trophy since winning the Spanish league in 2019. Koeman, a former Barcelona defender, returned to coach the club last summer after Barcelona finished last season without silverware for the first time since 2008.
Barcelona have now won the Spanish cup a record 31 times, including four straight from 2015-2018.
Bilbao have lost two Copa del Rey finals this month at La Cartuja Stadium in Seville. Bilbao lost 1-0 to Real Sociedad just two weeks ago when they met in the 2020 final that had been postponed due to the coronavirus pandemic.
Additional reporting by AP.