Still without a point this season, the hosts grabbed a shock lead when Atletico keeper Jan Oblak fumbled the ball over his own line on the stroke of half-time.
Getafe were reduced to 10 men in the 74th minute when Carlos Alena was sent off for a second bookable offence, and Suarez pounced to level four minutes later.
And the Uruguayan star struck in injury time when he headed home from a Sime Vrsaljko cross to seal the points.
Fellow veteran Radamel Falcao also struck late as Rayo Vallecano ended Atletico Bilbao's bright start to the season with a 2-1 win at San Mames.
Alvaro Garcia's opener for the visitors was cancelled out by a Pathe Ciss own goal, but substitute Falcao struck deep into injury time when he powered home a header from a Bebe free-kick.
Iago Aspas and Brais Mendez grabbed the goals as Celta Vigo claimed their first win of the season with a 2-0 success at Levante.
Riccardo Sottil put the hosts in front midway through the first half, but Matteo Darmian and Edin Dzeko struck in the space of four second-half minutes.
Robin Gosens and Davide Zappacosta scored early as Atalanta maintained their solid start with a 2-1 win over Sassuolo, who replied through Domenico Berardi, while Genoa twice came back from behind to draw 2-2 at Bologna.