Sami Hyypia's team, who replied through Sidney Sam, have lost four games in the last two weeks including a German Cup elimination at the hands of second tier Kaiserslautern and a 4-0 Champions League drubbing by Paris St Germain, both at home.
Leverkusen are on 43 points, 16 behind leaders Bayern Munich who visit Hanover 96 on Sunday.
Stuttgart's seventh successive defeat kept the five-times German champions fourth from bottom.
Nuremberg and Eintracht Braunschweig's basement battle provided all sorts of twists with three missed penalties and a 2-1 win for the hosts who scored both goals after being reduced to 10 men.
There was also drama at Borussia Moenchengladbach, where Hoffenheim came from two goals down to draw 2-2, and Freiburg, where Augsburg rallied to win 4-2 with three goals in the last 16 minutes.
Hamburg, the only ever-present team in the 51-year history of the Bundesliga, moved up to 16th, the relegation playoff spot, on 19 points, behind Stuttgart on goal difference.
Freiburg (18) slipped into the drop zone while Braunschweig remained rooted to the bottom on 15
Slomka, who inherited the worst defensive record in the league with 51 goals conceded, made six changes after replacing Dutchman Bert van Marwijk who was fired after Hamburg let in 20 goals while losing seven games on the trot.
Despite missing the injured Rafael van der Vaart, they looked more confident against a subdued Dortmund and went ahead three minutes before halftime when Petr Jiracek got in front of his marker to head in at the far post.
Pierre-Michel Lasogga added the second goal with a clinical finish after being sent clear by Tolgay Arslan before Calhanoglu spotted keeper Roman Weidenfeller off his line and scored with a quickly-taken free kick from near the centre circle.
"I hit the free kick the same way I do in training all the time. I'm delighted I pulled it off," the Turk told www.bundesliga.com.
"It's an absolute dream to be able to beat Dortmund. We're in a much better place now."
Stuttgart also looked set to end their losing streak when Ivorian midfielder Arthur Boka rifled home a 20-metre shot just before halftime to cancel out Levan Kobiashvili's early goal for the visitors.
Hertha substitute Wagner had other ideas as he headed an 87th-minute winner, although he was booked for his celebrations and sent off for another yellow card five minutes later.
"I feel I have absolute backing," said Stuttgart coach Thomas Schneider. "The work can be seen even if we are not getting the results."
The fun and games began in Nuremberg when the hosts had Per Nilsson sent off for bringing down Havard Nielsen in the 32nd minute.
Domi Kumbela headed Braunschweig in front two minutes later and missed a chance to extend their lead when he had a weak penalty saved by Raphael Schaefer five minutes before the break.
Hiroshi Kiyotake and Tomas Pekhart netted twice in as many minutes after halftime to put Nuremberg in front before the visitors missed another penalty when Ermin Bicakcic was also denied by Schaefer.
Five minutes later it was Nuremberg's turn when Marjan Petkovic touched Kiyotake's spot kick on to the post.
An exquisite lob by Patrick Herrmann gave Gladbach a fourth-minute lead against Hoffenheim.
Tony Jantschke headed the second goal 14 minutes later but the visitors fought back after the break through Roberto Firmino and a Sejad Salihovic penalty.