Villarreal had temporarily taken fourth place with a 2-1 win over Levante on Saturday after beating Real at the Bernabeu for the first time ever last week.
La Blaugrana shushed any voices that sprang out after the first defeat of the season against Espanyol with a thumping 5-0 victory over Real Betis.
"The players needed a game like this", Real coach Zinedine Zidane said.
As Ronaldo was led off the pitch to have his bleeding head quickly patched up he asked to see the phone of one of Real's medical staff and, pointing the camera at himself, used it to check the damage to his face.
Ex-Liverpool star Suarez found the net five minutes later with a terrific volleyed effort from inside of the area.
"We played well and Messi was incredible", Valverde said in his postgame news conference.
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Garcia saved one, but on the next, a strong forehand return opened up the court for the American who ripped a victor to go 4-2 ahead.
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What they said Tweet of the match Star man - James Tarkowski James Tarkowski's performace caught the eye (Martin Rickett/PA). If we score a second goal then you are in control but when they are alive, they are alive and they go until the end.
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Barca, the only unbeaten side in Europe's top five leagues, lead the way in La Liga with 54 points after capitalising on second-placed Atletico Madrid's 1-1 draw with Girona on Saturday and third-placed Valencia's 2-1 defeat at Las Palmas.
An end-to-end first 15 minutes saw both teams commit plenty of players in the final third of the field, but Betis were having the better moments, with Joaquin the next to miss Ter Stegen's post following a counter-attack.
A strike from Ivan Rakitic broke the home side's resistance in the 58th minute and top scorer Messi and the in-form Suarez took turns in adding to the scoring to cap a sixth successive league win for Ernesto Valverde's imperious team.
Before that, Barca are in action away to Betis and here, Goal takes a look at everything you need to know ahead of their trip to Seville on Sunday.
Faced with two defenders, Messi dummied to shoot, drifted past both and calmly slotted home past goalkeeper Antonio Adan.
Paulinho sustained an injury against Espanyol and is a doubt, while Ousmane Dembele is facing another spell on the sidelines, but Samuel Umtiti is pressing for a return from an injury which has kept him out for almost two months.
There was still time for Suarez to complete his brace with a crisp finish after Messi had driven at the heart of an over-matched defence.