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."
Second-placed Gladbach, also on 13 points, sunk struggling Hamburg when Igor de Camargo headed in a free-kick on 66 minutes.
Last-placed Hamburg have yet to win this season and stretched their losing streak to four straight league games, piling further pressure on coach Michael Oenning.
"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.
Cologne moved up to 13th after Lukas Podolski sparkled in their 4-1 win at Bayer Leverkusen, scoring twice and setting up another for Milivoje Novakovic. Mato Jajalo completed an unexpectedly easy win in stoppage-time.
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.
Hoffenheim continued their fine season under coach Holger Stanislawski with Roberto Firmino scoring twice and Ryan Babbel adding another in their win over Wolfsburg, who dropped to 14th.
