But the Barcelona's previous record was at 31 league games unbeaten under former manager Pep Guardiola back in 2010-11 season.
The reigning champions' title hopes are all but over, with Real 14 points adrift of Barca, but a run of four straight wins has ended any worries of missing out on a Champions League spot.
Their lead was short-lived as Messi set up Suarez to equalise.
He mesmerized by scoring a Ronaldinho-esque free kick that slid under the jumping Girona wall and past the right hand of the goalkeeper. Just when you think you've seen everything from Messi, he surprises you, like his free kick, which looked easy but was very hard to execute.
"But then he moved to the Galaxy, so Philippe had to take responsibility himself and he put the team on his back". Then when you see it happen you think 'ah, of course.' Leo makes it look easy, but it isn't easy.
It was the first time Barcelona has started with Messi, Suarez, Coutinho and Ousmane Dembele on the pitch as they demonstrated just how devastating they can be in attack.
National Basketball Association agent Dan Fegan dies in Colorado auto crash
Fegan had represented many National Basketball Association players, including John Wall , Demarcus Cousins and Dwight Howard . Dallas Mavericks owner Mark Cuban remembered Fegan as a man with a good heart.
French winger Dembele played his part too late on as he crossed from the left for Suarez to complete a hat-trick. Portu's strike from an angle hit both posts before settling into the net.
File image of Barcelona's Nelson Semedo.
Girona fell into ninth place.
And Coutinho got in on the act in the 66th minute with a brilliant Barca fifth from 25 yards.
Cristiano Ronaldo and Real are finding their form at the flawless time, and that should concern a lot of teams in Europe. The Frenchman made no mistake, converting from the spot in the 89th minute. But who had the better goal?
It was the first real glimpse of what Barcelona's front three are truly capable of when they are all on top form. It was a nice gesture that Ronaldo allowed Benzema to take the penalty.