But Jose Pozo levelled for the hosts just before half-time and Alvaro Garcia gave them the advantage 12 minutes after the pause.
But just as it appeared that Ernesto Valverde’s men were about to suffer their second defeat of the season, second half substitute Ousmane Dembele equalised.
A stunned Vallecano were then hit two minutes later in the 89th minutte when Suarez bagged his brace.
"Suarez has been getting better and better for a long time," Valverde said. "Now he's getting more goals and now he's scoring them when we also need them most. We are delighted."
The last gasp victory, coupled with Atletico Madrid’s 1-1 draw with Leganes, allowed Barca to move four points clear at the top. However, that gap could be reduced to one point should Deportivo Alaves win at Eibar on Sunday afternoon.