Saul and Griezmann scored at BayArena before a Kevin Gameiro penalty and late Fernando Torres goal.
Griezmann was delighted to surpass Luis Aragones as Atletico's top goalscorer in the Champions League, but he wants more.
"We knew it was going to be a tough game," he told Spanish TV, via UEFA.com.
"We wanted to win, we did that and the return leg will be difficult as tonight was.
"I'm proud to become part of Atletico's history with my goal and hopefully I can carry on making history here.
"They pressured us and made life difficult, but the boss's changes helped us win the game.
"We have a big game to play now in front of our own fans."
The second leg of the last-16 tie is at the Vicente Calderon on March 15.