A goal from free-kick specialist Vincenzo Grifo had put Freiburg into a first-half lead and on course for a valuable three points in their top-four bid, but Azmoun's first goal in Leverkusen colors meant the honors were shared.
Grifo opened the scoring with his 12th league goal of the season.
Bayer manager Xabi Alonso brought on midfielder Florian Wirtz at half-time and the change had the desired impact, with the visitor benefiting from the teenager's creativity and vision.
With 67 minutes gone, Wirtz moved the ball to Mitchell Bakker, who chipped a superb looping cross to Azmoun. The Iranian cut the ball across the face of goal and into the net.