Ernesto Valverde's men frustrated their hosts for long stretches at Camp Nou and they appeared set to steal a well-earned point until the final 12 minutes.
The goals came courtesy of Neymar, but Messi played the role of provider on both occasions.
Barca's inspirational Argentinian first threaded a fine throughball for the Brazilian and then dazzled with a clever run and pass for Neymar's second in the 85th minute.
After the match, both Luis Enrique and Neymar were left in awe of Messi's ability, referring to him as the best player in the world.
AS quoted Neymar as saying: "Messi is the best in the world, a star. I am very happy with the game today. It went perfectly with two goals.
"We knew we were going to suffer a little but we managed to keep defending well and then get the win."
Luis Enrique acknowledged that if his side is to get close to the standards once set by Pep Guardiola's Barca, they will need to strike the right balance between attacking and defending.
However, he was content with what he saw on Saturday as his side refused to give up and snatched the win late on.
He added: "To be a team that wins titles you must have a good attack and a defensive phase, to be a complete team.
"Sometimes you can play too long and others you can have possession. Getting the right balance isn't always easy.
"A team is formed to unite different personalities and profiles of players. I'm happy with what I have.
"I won't explain why whoever plays to the players and I won't do it here. We always hope to have the best version of each player.
"Any individual thing is easy to achieve but overall, the coordination of everything, means many hours of training. We do what we can, with and without the ball."
