Messi opened the scoring and then helped tee up Neymar and Paco Alcacer to lead Barca to a record 29th Cup triumph after Theo Hernandez’s brilliant free kick had briefly brought Alaves level.
Victory means Enrique bows out after three seasons in charge with nine trophies to his name, including the Cup in every year of his reign.
However, once again it was Messi who inspired Barca to bounce back from the disappointment of losing their La Liga title to Real Madrid last weekend with a stunning individual display.
“He is extraordinary, an extraterrestrial,” said Enrique of Messi. “I have had the opportunity to enjoy the best of Messi, at his peak athletic performance.
“He is the number one as he shows in his performances, and it has been a true pleasure.”
Alaves boss Mauricio Pellegrino was equally in awe of the five-time World Player of the Year.
“He is an out of the ordinary player,” said Pellegrino. “He has the ability to create goals as well as score them with the precision of his passing.”
Barca were without the suspended Luis Suarez and Sergi Roberto, and Enrique was forced into another reshuffle just 11 minutes in as Javier Mascherano was forced off with a head injury.
Yet, while Barca’s depth has been much-criticised this season, Andre Gomes deputised ably at right back after replacing Mascherano and Alcacer took his chance on a rare start on the big stage in Suarez’s absence.
In stark contrast to Barca’s glorious run of six finals in seven years, Alaves were appearing in Spanish football’s showpiece for the first time.
The Basques were roared on by the more enthusiastic support at a far from full Vicente Calderon in its last competitive fixture before Atletico
And Alaves were left wondering what might have been had they gone ahead when Ibai Gomez’s effort came back off the post, ricocheted off the back of Barca goalkeeper Jasper Cillessen, but somehow squirmed along the goal line and not into the net on 27 minutes.
Three minutes later, Messi kicked off a thrilling four-goal spell in the final 15 minutes of the half as he exchanged passes with Neymar before curling his 54th goal of the season.
However, Barca’s initial lead only lasted three minutes.
On-loan Atletico left back Hernandez has been courted by
And the 19-year-old showed why he is in such demand by bending a sumptuous free-kick into the top corner.
Messi saw a free kick of his own beaten away by Fernando Pacheco, whilst Ivan Rakitic drilled a low shot just wide as Barca immediately reacted to conceding.
Fittingly, Messi was involved once more when Barca did regain the lead as he received Neymar’s pass before playing in Gomes, and the Portuguese’s low cross was tapped home by Neymar at the back post as he scored for the third consecutive season in the Cup final.
Barca then put the game to bed in first-half stoppage time when Messi again pulled the Alaves defence apart, and his reverse pass freed Alcacer to smash the ball beyond Pacheco into the far corner.