Messi opened the scoring in Barca's 3-1 win over Alaves in Saturday's final to finish the season with 54 goals in all competitions.
Theo Hernandez cancelled out that goal, but Neymar and Paco Alcacer struck late in the half as Barca claimed the Copa for the third straight year.
Luis Enrique was full of praise for Messi afterwards, saying the Argentina star was from another planet.
"Messi is an extraterrestrial in every sense and I'm very fortunate that I was able to enjoy the best, or one of the best, versions of him," he said.
"Without a doubt he's the number one and to be that you have to control every aspect and be very strong physically.
"He takes perfect care of himself, and there's still a lot more to come from Messi."
The trophy marked a ninth for Luis Enrique since he took charge of Barca in 2014.
The 47-year-old said he would be leaving the club without sadness after a successful spell at the helm.
"I feel happy because I have enjoyed a lot making Barca fans happy by winning trophies and that makes me proud," Luis Enrique said.
"I am the one who has decided to stop because it will be the best for me and the for the players, they need potent and intense stimulus.
"Nine trophies out of 13 is a pretty good record."