Robert Lewandowski had a good chance to put Dortmund back ahead on the half-hour mark when he tried to clip over Manuel Neuer, but the Bayern goalkeeper spread himself just enough to make the save.
Then just before the half-time interval, Gomez saw his effort saved by Roman Weidenfeller and Neuer was called into action moments before the hour mark by a Lewandowski effort.
The Polish striker should have scored minutes later when he stepped up to take the penalty awarded to Jurgen Klopp's side following Jerome Boateng's handball offence inside the Bayern area. However, he saw his effort from 12 yards turned away by Neuer.
Four minutes later Rafinha was dismissed for a second bookable offence as the visitors were reduced to 10 men.
Dortmund failed to take advantage of their numerical superiority and had to settle for a 1-1 draw.