Messi missed a league game for the first time this season to attend the birth of his third son Ciro, but Barca barely needed their talisman against free-falling Malaga, who had striker Samu Garcia sent off after half an hour for a studs- up challenge on Jordi Alba.
Suarez headed home a looping cross from Alba to give Barca the lead in the 15th minute and his former Liverpool team mate Coutinho struck the second in the 28th with a backheel from close range after an impressive dribble from Ousmane Dembele.
Malaga were the last team to beat Barcelona in the league, derailing their title bid with a 2-0 win last April and they created two chances in the first half but could not test visiting goalkeeper Marc-Andre ter Stegen.
Messi will return when Barca host Chelsea in a Champions League last-16 second leg game on Wednesday but captain Andres Iniesta is fighting to return from a hamstring injury.
“Leo is the best player in the world and we always need him,” Coutinho told reporters.
“We congratulate him and we are looking forward to having him back in the next game.
“We came here expecting a difficult night but we managed to score early and with an extra man we were able to control the game.”
Barca are unbeaten in the league and lead the standings with 72 points, 11 clear of Atletico Madrid who host Celta Vigo on Sunday. Real are third on 57 after seeing off dogged Eibar with two strikes from Cristiano Ronaldo.
The prolific Portuguese scored for the seventh successive game in all competitions by blasting into the net in the 34th minute after chesting down a long ball by Luka Modric. Ivan Ramis pulled one back for Eibar when he outjumped Sergio Ramos to head home from a corner in the 50th.
The irrepressible Ronaldo had the final say, however, meeting a cross from Dani Carvajal in the area and powering the ball beyond the Serbian goalkeeper in the 84th minute.
Ronaldo has roared back from a feeble start to the season, when he scored just once in his first eight league games, to reach 18 goals in the campaign.
“He’s from another galaxy,” Real coach Zinedine Zidane said of Ronaldo.
“He knows he’s going to score, he’s unique and special. His stats speak for him. And he’s our player. He’s always positive and if he scores twice, he wants to score a third.”
Valencia tightened their grip on fourth place and took a giant leap towards returning to the Champions League by winning 2-0 at Sevilla, opening up an 11-point gap on the Andalusians in fifth place.
Spain international Rodrigo Moreno scored either side of halftime to inflict a third home defeat in little more than two months on Sevilla, whose best chance of getting back into Europe’s elite competition now likely rests on them winning the tournament.
Vincenzo Montella’s side visit Manchester United on Tuesday in a last-16 second leg with the tie finely balanced at 0-0.
Struggling Levante snapped a run of 15 games without a win by beating Getafe 1-0 away from home in coach Paco Lopez’s first game in charge, lifting his side four points above the relegation zone.
Meanwhile, Real Madrid captain Sergio Ramos took an unexpected pitstop during his side’s La Liga game against Eibar on Saturday, running from the pitch and down the tunnel to go for an emergency toilet break.
Ramos spoke to referee Jose Luis Munuera Montero in the 73rd minute of the game then darted off to the visiting dressing room, with Real match delegate Miguel Noguera, known as Chendo, going in to check on the player after a few minutes.
The Spain international returned to the pitch in the 78th minute, shortly before Cristiano Ronaldo struck the winning goal in a 2-1 win for Zinedine Zidane’s side.
The French coach started laughing when he was asked about the incident in the post-match news conference and said “he went for a shit.”
Ramos poked fun at himself after the game on social media, tweeting “nature calls” along with a picture of him and Ronaldo celebrating the goal.