Die Borussien, who were beaten by Hamburg last week and lost to Krasnodar in the Europa League on Thursday, looked in no mood to endure another defeat and dominated possession from the start at Signal Iduna Park.
However, the Polish keeper was beaten moments later as Gonzalo Castro nodded the ball into the net.
Castro then turned provider on 19 minutes when he teed up Pierre-Emerick Aubameyang rifled the ball high into the roof of Tyton's goal.
Mkhitaryan was again denied by Tyton late in the half as Dortmund went in search of a third goal to try and seal victory before half time.
However, Didavi cropped up with five minutes of the half left to score for Jurgen Kramny's men and give VFB hope of a comeback.