Enrique is leaving after three years in charge at the Nou Camp.
Ibai Gomez hit the post for Alaves before Messi scored his 54th goal of the season with a 20-yard curler.
Theo Hernandez bent in a free-kick to equalise, but Neymar turned in from close range and then Messi played in Paco Alcacer right on half-time.
It has been a disappointing season for Barcelona, who finished runners-up to Real Madrid in La Liga and were eliminated from the Champions League by Juventus in the quarter-finals.
They did at least restore some pride with victory over last season's second division champions in what was the final match to be played at Atletico Madrid's Vicente Calderon Stadium.
There were some records set along the way as Enrique's team retained the trophy, having beaten Athletic Bilbao and Sevilla in the last two finals.
Messi, who struck the opener after exchanging passes with Neymar outside the penalty area, became the first player to score in four Copa finals since Athletic Bilbao's Telmo Zarra in 1950.
Neymar, who forced in Barcelona's second when Messi played in Andre Gomes to cross low from the right, is the first man to score in three successive Copa finals since Ferenc Puskas, who achieved the feat with Real Madrid between 1960 and 1962.
Messi's through pass for the third goal was sublime, and was finished calmly by Alcacer - who has been inconsistent since his move from Valencia last summer.
The one major worry for Barcelona was a nasty head injury to Javier Mascherano, who was carried off on a stretcher after an early collision with Marcos Llorente,
He suffered concussion as well as a cut to his head, and an injury to his right knee that will need further tests, the club said afterwards.
Lionel Messi was inspired as Barcelona beat Alaves to secure the Copa del Rey for the third year running and give coach Luis Enrique a winning send-off.