Lionel Messi then bagged a quickfire brace before Suarez completed his hat-trick, with Phillipe Coutinho also getting onto the scoresheet with a stunning effort from range. He's also set to miss league fixtures against Atletico Madrid, Las Palmas and Malaga, along with Espanyol in the Copa Del Rey.
Barca's run surpasses their previous club record of 31 games in a row without defeat, which was set back in 2010-11, when Pep Guardiola's side won 27 games and drew four between matchday three and 33 of the season.
However, Barcelona didn't have matters al their own way in this fixture with Girona taking a shock lead, after just three minutes, when Portu scored.
There was enough time for Barca to grab a fourth goal before the break when Coutinho and Messi and combined to set up for Suarez for his second of the match.
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Luis Suarez hit a hat-trick and Philippe Coutinho scored a brilliant first league goal for Barcelona as the La Liga leaders thrashed Girona 6-1 on Saturday to open up a 10-point lead at the top of the table.
Messi nearly put Barcelona ahead after 19 minutes after Suarez had picked him out in the box.
"Just when a player seems to have done everything, he goes and surprises you". You see it and you think it's easy.
"Leo is incredible", Coutinho told Movistar.
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But it took him five games in one of the best sides in the world to bag in La Liga.
He added another goal six minutes later with a superbly struck free kick that he sent low under the players in the defensive barrier when they jumped, anticipating he would try to shoot over them.
It was the first time Messi, Suarez, Coutinho and Ousmane Dembele were named in the same starting XI by Ernesto Valverde, but there is little doubt over who remains the master. "[Messi's] very important for us".
"Dembele offers more width, he's a winger and he's still very young".
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Nelson Semedo, meanwhile, will undergo tests on a hamstring injury on Sunday after being forced off in the final minutes of the game.