The Argentina forward completed his 38th Barcelona hat-trick when he finished off from another Alba cross.
Dembele could make an impact from the bench as Barca look to take advantage of Real Madrid's failure to beat Levante earlier on Saturday.
The lead was doubled on 35 minutes as Messi grabbed his second goal by turning home Gerard Deulofeu's low cross.
Suarez who shrugged off an injury whilst on global break missed some decent chances, the closest being in the 55th minute when he blazed wide in the six yard box after a cutback from Messi. "Game by game he keeps showing that he will be around for a long time to come".
Speaking on his new signing, Valverde said: "Dembele has been getting better every day, [but] he is a very young 20-year-old kid".
Suarez's goal was met with chants for club president Josep Maria Bartomeu to resign.
The fixture closed out with the hosts continuing to control the game, as they had done all evening, and their dominance again shone through with five minutes to go as Pique rose above all else inside the area to convert Rakitic's fiercely-whipped corner.
"We can't be happy with our performance", said Real manager Zinedine Zidane.
With a squad that looks weaker than last season and most importantly, weaker than what Real Madrid have to offer, this season was always going to be about Messi, first and foremost.
Ernesto Valverde's men now have a four-point lead over their rivals and Suarez feels each point is important as the title race can get close.
Sevilla moved up to second, just two points behind the leaders, as goals from Paulo Henrique Ganso, Wissam Ben Yedder and Nolito secured a 3-0 win over Eibar.