The Brazil global netted his first Barca goal with six minutes remaining as Ernesto Valverde's men recovered from a terrible first half to claim a fourth straight top-flight victory in 2017-18.
But two substitutes, Suarez and Paulinho, both struck after the break as Ernesto Valverde's men continued to set the pace at the top of the table.
Barca's fine start to the season means they already sit seven points clear of reigning champions Real Madrid, whose game in hand is a testing away trip to in-form Real Sociedad on Sunday.
In the 16th minute Getafe squandered a great opportunity to grab the lead after Molina failed to connect to Baku Shibasaki's deflected cross.
"We have shown that we can come back", he said.
Dembele, making his first start in La Liga, fell to the ground clutching his hamstring after sprinting for the ball just after the 20-minute mark.
Messi nearly found a quick-fire equaliser, his free kick curling towards Guaita's top-left corner, but the goalkeeper managed to tip it over.
Former Tottenham midfielder Paulinho scored a late victor as Barcelona maintained their 100% record in La Liga by coming from behind to beat Getafe.
Both teams continued to press for a victor but it was the visitors who managed to strike the decisive blow when Paulinho, who was brought on in the second half in place of Ivan Rakitic, drove into the box before firing a lovely finish across the keeper. Fellow new recruit Paulinho then became the unlikely hero with a goal that issued a considerable statement to his doubters.