Any doubts over Enrique's job in the short term were blown away by arguably his side's best performance in his seven-month reign.
Messi teed up Neymar to slot home the opener after 12 minutes and then repeated the trick to allow Luis Suarez to register just his second La Liga goal.
The Argentine inadvertently handed Atletico a route back into the game when deemed to have brought down Jesus Gamez inside the area on the hour mark and Mario Mandzukic slammed home from the penalty spot.
Messi made amends when he slotted home the clinching goal from close range three minutes from time to move Barca back to within a point of Real Madrid at the top of La Liga.
"This win has a special value because it is against a direct rival for the league," Suarez said.
"At the moment the self-confidence in the team is good. This win will gives us a big boost because it came against Atletico Madrid.
"It is spectacular to play with Messi and Neymar, just like the rest of my team-mates. We have achieved a great victory.