The midfielder was left unconscious following a first-half collision with team-mate Cristian Ansaldi, who was challenging Wolfsburg's Ivica Olic for an aerial ball.
Suarez required lengthy treatment on the pitch before being replaced by Tiago, with Atletico later confirming details of the injury.
"Mario Suarez suffers a traumatic brain injury," they tweeted. "He's regained consciousness. He's being taken to the hospital for tests."
Diego Simeone's men started well and Mandzukic tested goalkeeper Diego Benaglio inside the opening minute as the Liga champions looked to end a run of three matches without a goal.
And it was not long before they were in front, with Garcia rising well to power home a header from Gabi's corner in the 13th minute.
And the visitors were three up eight minutes later when Mandzukic was clumsily brought down in the penalty area by Naldo - the referee showing no hesitation in pointing to the spot.
Naldo atoned for his earlier mishap by drilling into the bottom left-hand corner from a 67th-minute free-kick, but Koke's curling effort ensured the German club's joy was short-lived.
Despite the serious injury to Suarez, who later underwent a CT scan and was kept in hospital for observation, the nature of the result and a debut for Griezmann should send the men from the Spanish capital home with at least some cheer.
