Luis Suarez bagged a double for Barcelona but their La Liga victory came at a cost as Messi was ruled out for up to two months after damaging knee ligaments.
Villarreal's victory came courtesy of a goal from Leo Baptistao, who finished clinically having been found by Roberto Soldado inside the area after 14 minutes to secure their fifth straight win.
The best chances for a below-par Atletico fell to Antoine Griezmann at the end of each half, but he was denied by Villarreal keeper Alphonse Areola.
Villarreal are surprise leaders with 16 points from six games, a point more than Barca and two ahead of Real.
Iago Aspas grabbed a 76th-minute equaliser to give fourth-placed Celta Vigo a 1-1 draw at Eibar.
Barca, who were looking to rebound from a 4-1 hammering by Celta Vigo on Wednesday, lost Messi after just nine minutes following a clash of legs.
The Argentine, who was in the process of shooting, collided with Daniel Castellano after three minutes.
He tried to continue but was eventually forced off the pitch, with the club later confirming that tests had shown he had suffered knee ligament damage.
BEST PLAYER
“He (Messi) is the best player in the world, our best player and the most dangerous so we are going to notice this,” midfielder Sergio Busquets told reporters.
“We are having a lot of bad luck with injuries and particularly now when we cannot sign players.”
Suarez gave Barca the breakthrough with a powerful header into the corner from a Sergi Roberto cross after 25 minutes.
The tension was taken out of the game when Suarez doubled his tally, cracking the ball into the roof of the net from 12 metres after Munir El Haddadi crossed from the right and Busquets left the ball for the former Liverpool striker to finish.
Neymar blazed a penalty over the crossbar, following a handball by Antolin Alcaraz, after 66 minutes and Las Palmas pulled a goal back with a deflected Jonathan Viera strike two minutes from fulltime.
Real pushed forward from the start in their match against Malaga but a combination of fine goalkeeping from Carlos Kameni and near misses prevented them from breaking the deadlock.
Malaga’s most dangerous player Nordin Amrabat was sent off for elbowing Marcelo after 76 minutes, but Real could not find a way through, and it was a frustrating match for Cristiano Ronaldo and his fellow forwards.
"We are angry because we could have won and didn’t,” Real keeper Keylor Navas told reporters.
“There’s a long way to go, we will try not to lose more and achieve our objectives.”
Isco, who was lively for Real and stretched the Malaga defence, had the best chance of the first half when he volleyed a deep Marcelo cross just wide of the post.
Kameni was a rock at the back making a series of saves including a block from a Ronaldo volley after the break that left the Portuguese looking skywards.
When Kameni was finally beaten by an Isco shot, Weligton was behind him to clear the ball off the line.
Amrabat was red-carded for throwing an arm at Marcelo as he attempted to scamper clear on the counter, but even with a man extra, Real’s luck did not change.
(Reporting by Tim Hanlon, editing by Toby Davis)
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