Junior Firpo also scored for Barcelona, which moved within a point of league leader Real Madrid, which plays at Atletico Madrid later Saturday.
- Barcelona ratings: Ter Stegen, Firpo rise in road win
Barcelona played with 10 men from the 82nd minute as defender Clement Lenglet was sent off with a second yellow card for a hard foul on an opponent.
Ter Stegen set up Suarez's opening goal in the 41st after leaving his area to stop a breakaway. He controlled the ball with his chest before sending it the other way to leave Suárez in a one-on-one situation against Getafe goalkeeper David Soria. The Uruguay striker calmly one-touched a lob shot over Soria for his third goal of the season.
"Let's not fool ourselves, I saw him, but not that clearly," Ter Stegen said, with a smile. "It was the perfect timing and Luis took advantage of it. It was a wonderful goal."
Barcelona manager Ernesto Valverde also praised his German goalkeeper for his footwork.
He's incredible with his feet. It's not the first time he's set up a goal,"Valverde said. "He gives us so much in many senses. But the assist is just an anecdote, we'll prefer it if he keeps on making saves."
Firpo scored his first goal for Barcelona from close range in the 49th after being set up by Carles Perez, Messi's substitute in the attack. It was the first away win for Barcelona, and its first league game without conceding. Messi was out again after injuring his left abductor muscle in the first half of Barcelona's 2-1 win against Villarreal on Tuesday, his first start of the season following a long layoff because of a right calf injury sustained preseason.
Valverde also could not count on Ousmane Dembele because of a muscle ailment.
Getafe, which dropped to 11th place, had been unbeaten since a loss at Atlético Madrid in the league opener.