Barca edge out a victory at La Rosaleda
Atletico Madrid, meanwhile, delivered a 5-0 thrashing of Rayo Vallecano, extending the team's strong start to the Spanish league.
Barcelona kept Bernd Schuster's team pushed back in its area early on at La Rosaleda Stadium but failed to finish its scoring chances until Adriano broke through in the 44th minute.
Gerard Pique used his back to deflect the ball off the crossbar for Barcelona in the 58th before Malaga finally dared to look for an equaliser it nearly found through 17-year-old Fabrice Olinga's shot from close to the penalty spot that hit the post in the 68th.
"We went from better to worse," said Barcelona coach Gerardo Martino. "Our first half was good, with precise, dynamic ball movement and good play on the wings, but we lacked effectiveness and scoring chances. We needed to be more consistent (in the second half). We have to finish games off and not leave the rival the possibility to come back."
Neymar went on as a substitute for the third straight game for Barcelona in the second half and shone with his dribbling, pace and a pair of shots that almost doubled the final margin.
Martino said that after his performance, Neymar looks prepared to get his first start.
"There is a difference between what we saw from Neymar today and the other day against Levante," Martino said. "I don't know how long it will be before he is starting, but in his head Neymar is ready."
Messi bruised a muscle in his left thigh during Wednesday's 1-1 draw at Atletico Madrid in the first leg of the Spanish Super Cup.
While Barcelona may still be without its all-time leading scorer for this Wednesday's return leg at Camp Nou, Atletico warmed up by thrashing Rayo Vallecano 5-0 earlier on Sunday for its second victory in as many rounds.
Diego Simeone's Atletico side dominated the Spanish capital derby for Atletico's second win in as many rounds before it faces the return leg of the Spanish Super Cup at Barcelona on Wednesday with the tie poised 1-1.
Turan passed for Diego Costa to make it 2-0 in the 20th before then scoring himself from a pass by Diego Costa in the 35th.
Turan crossed for Tiago Cardoso to head in the hosts' fourth goal in the 53rd before Garcia got his double in the 90th.
Also, 10-man Celta Vigo emerged victorious 2-1 from a wild match at Real Betis, while Sevilla drew 0-0 at Levante as both teams remained winless.
Cesc Fabregas was tasked to lead Barcelona's attack in place of Messi, and it took him 15 minutes to start causing havoc in Malaga's defense.
The former Arsenal midfielder's best piece of playmaking came in the 19th when he threaded a through ball to Pedro Rodriguez, only for the forward to lose control while trying to dribble goalkeeper Willy Caballero.
"I always try to play the same way, with motivation and working to improve," Fabregas said. "I know this is an important season for me. Our fans have asked more from me and I am trying."
Caballero turned back Alexis Sanchez's point-blank header in the 37th, but Adriano surprised him with a blistering left-footed effort from outside the area that the Argentine got an arm on but couldn't keep out.
"There is always something you could have done when they score on you, but by then it's always too late," Caballero said.
Adriano was substituted by Dani Alves at halftime. The club said it was "for precaution" after the Brazilian defender felt "pain in his right leg."
Neymar, who scored Barcelona's goal at Atletico on Wednesday, replaced Pedro in the 63rd and immediately showed his speed and ability to break down defenders one-on-one. His dribble between the legs of Jesus Gamez earned him a shove as the humiliated defender let his anger get the best of him.
Neymar's footwork produced a foul on the edge of the area that he turned into a curling free kick which Caballero dove to parry in the 80th.
And seconds after Valdes saved a last-gasp header by Sebastian Fernandez in the 90th, Neymar pounced on a loose ball in the area, but his low shot clipped a defender's boot and looped over the bar.
"The plan was to arrive alive to the last stage of the game and then fight for a good result," said Schuster. "The only thing that went wrong was that luck wasn't on our side. If you have three chances, you have to put one in."
After Betis and Celta had both rattled the woodwork, Manuel "Nolito" Agudo guided Celta to a 2-0 lead by assisting Charles for a headed goal in the 67th before firing in a goal of his own in the 74th.
But Celta's Borja Oubina received his second booking with five minutes to play. Betis pulled one back through Ruben Castro in the 90th and grazed the draw when Manuel "Chuli" Vazquez volleyed off the goal frame in injury time.
Real Madrid visits Granada on Monday amid intense speculation that it will close the transfer of Tottenham winger Gareth Bale.
Information from The Associated Press was used in this report.