Earlier, Cristiano Ronaldo rounded off a week where he was named World Player of the Year for the third time by scoring twice as Madrid struck three times in the final half hour to beat Getafe 3-0 yesterday.
Atletico Madrid, who dumped Real out of the Copa del Rey in midweek, moved back to within four points of their city rivals in third as goals from Mario Mandzukic and Raul Garcia handed them a 2-0 win over managerless Granada.
"He hasn't dipped at any point. He has been world class."
In a change from his much criticised rotation policy, Enrique named the same side that beat Atletico 3-1 at home last weekend and was rewarded with another bright display illuminated by Messi's class in front of goal.
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The Argentine's first effort of the game was brilliantly saved by Fabricio after a well-worked corner, but the Deportivo goalkeeper was helpless when the four-time World Player of the Year opened the scoring after 11 minutes with a thunderous header from Ivan Rakitic's cross.
However, by contrast, Luis Suarez's troubles in front of goal continued as he fluffed his lines twice in the first-half with just Fabricio to beat.
Messi showed the Uruguayan how it should be done once more when he rounded off his 30th hat-trick for the club after the break when he cut inside onto his favoured left foot and fired into the far corner before an own goal from Deportivo defender Sidnei rounded off the scoring.