Messi opened the scoring 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 brough 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 bounceback from the disappointment of losing their La Liga title to Real Madrid last weekend with a stunning individual display.
Yet, whilst 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.
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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 Madrid move home next season.
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 Barcelona, but looks set to become the first player to cross the Madrid divide since 2000 by signing for Real Madrid in the coming weeks.
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.
Alaves battled bravely as the second period wore on, but any hope of a fightback was extinguished by the offside flag when Deyverson fired home from close range 20 minutes from time.