Messi paraded his record fifth Ballon d'Or to the Barca fans before kick-off and opened the scoring from the penalty spot after Gorka Iraizoz was sent-off just three minutes in for bringing down Luis Suarez inside the area.
Neymar made it 2-0 before the break when Messi was replaced by Arda Turan.
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The Argentine's presence wasn't missed, though, as Suarez hit a second-half hat-trick and Ivan Rakitic also tapped home after brilliant play by Neymar to keep Barca just two points behind league leaders Atletico Madrid with a game in hand.
"We have to wait and see what the tests say, but it was a slight strain and we preferred not to risk it," said Enrique.
Barca face Athletic once more in the first leg of their Copa del Rey quarter-final tie on Wednesday.
Should the holders get through they will have played 17 games in January and February alone, having already overcome a busy first half of the season due to Spanish Supercup, UEFA Supercup and Club World Cup commitments.
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"We spoke and it didn't make any sense to risk it at 2-0 and with all that we have to come," added Enrique.
"If I don't like taking risks with any player, imagine what I am like with Leo Messi."
Athletic denied Barca a clean sweep of six trophies in 2015 on their last trip to the Camp Nou as a 1-1 draw sealed a 5-1 aggregate win for the Basques in the Spanish Supercup in August.
However, any chance the visitors had of frustrating Barca once more was ended when Suarez outmuscled Borja Exteita and was brought down by Iraizoz after rounding the 'keeper.
"What happened in the game wasn't anything like what I or (Athletic coach) Ernesto Valverde would have imagined," Enrique continued.
"From that incident in the first few minutes the game changed completely with us in front and them down to 10 men.
"It was a much more comfortable game, but we had to work hard to create chances. In the second-half we did that very well."
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