The win took Atletico top of the standings on 14 points from six games while Getafe are bottom with no points.
The home side went in front on the stroke of halftime with a header from Stefan Mitrovic after Atletico keeper Jan Oblak had uncharacteristically spilled a cross.
Getafe played the final phase of the match with 10 men after Carles Alena was shown a straight red card in the 74th minute for raking his studs down the back of the leg of Atletico substitute Matheus Cunha.
The sending off galvanised the visitors and it took a superb double save from David Soria to prevent Angel Correa and then Suarez from equalising.
But the 10 men could only hold on for so long and Atletico levelled when Suarez controlled a cross from substitute Mario Hermoso with his right foot before blasting the ball into the net with his left in the 78th minute.
The Uruguayan then snatched the winner in added time, ghosting away from his marker Mitrovic to head in a cross from Sime Vrsaljko in added time.