Barcelona scored two second half goals in four minutes as Ernesto Valverde's side came from behind to beat Real Sociedad at Anoeta in San Sebastian.
The home side took the lead in 12th minute through Aritz Elustondo but strikes from Suarez and Dembele ensured Barcelona maintained their perfect start to league season, keeping their spot at the top of the league.
Key Talking Point
While Barcelona boast a wealth of attacking options, their defence has looked shaky in the past and that was on display again on Saturday when they went a goal behind to La Real - for the sixth league game in a row at Anoeta.
Gerard Pique is past the peak of his powers and needs greater support from his midfielders, while Samuel Umtiti was booked as the game ticked towards a close.
Sure, Barca can usually count on their attack to bail them out when they get sloppy at the back, but it's never great to be overly reliant on that firepower.
Starting XI: Ter Stegen (7), Semedo (6), Umtiti (7), Pique (6), Alba (7), Roberto (6), Rafinha (6), Rakitic (7), Messi (7), Suarez (7), Dembele (9*).
Substitutes: Coutinho (7), Busquets (7), Vidal (6).
STAR MAN - Ousmane Dembele was absolutely electric on Saturday and deserved his goal, a scrappy central finish following a corner.
The World Cup winner has started the season brilliantly and his strike against La Real was his his fourth in all competitions.
WORST PLAYER - Ruben Semedo had a bit of a shocker at right back and was substituted for Philippe Coutinho as Barca chased the game.
Sergi Roberto moved to right back and the away side instantly looked more comfortable, with Coutinho giving his team a bigger attacking threat than before.
Barcelona now turn their attention to the Champions League, with a home tie against PSV coming up on Tuesday.
Their next La Liga game is a Camp Nou affair against Girona.