Wiese was dismissed in the 17th minute for a foul but Bremen went ahead seven minutes later after scorer Mehmet Ekici combined well with Claudio Pizarro to curl in from 14 metres.
Philipp Wollscheid levelled for the hosts on the hour after the start of the second half had been delayed by heavy rain and hail for 20 minutes.
"It was certainly not boring," Bremen coach Thomas Schaaf told reporters. "We made it hard on ourselves with the red card and we put ourselves in a difficult position.
"But we scored and after our goal we remained disciplined so I think the draw was deserved."
"I must admit that Gladbach's win was deserved. We again made a mistake that led to the goal and such mistakes have been accompanying us for weeks," said Oenning.
Podolski, whose Germany spot has been challenged by Leverkusen's Andre Schuerrle in recent months, silenced his critics with a stellar performance as rival Schuerrle was dismissed for a harsh foul in injury-time.
