Ken Reichel gave the visitors the lead early in the second half before a penalty from Stefan Kiessling saved the hosts from an even worse result.
They nevertheless saw their advantage over fifth-placed Wolfsburg reduce to a single point, jeopardising their hopes of qualifying for the Champions League next season, and drawing derision from their own fans.
Leverkusen had ended their nine-game winless streak in midweek, but that did not seem to have settled them particularly much.
Faced with a side fighting for their lives at the foot of the table, they found it hard to break their opponents down, succeeding only once in the first half with Andres Guardado's early shot missing the target.
It was an otherwise predominantly scrappy affair, the kind of game which suited Braunschweig more than their hosts, and they took the lead early in the second half with their first effort of the game.
Mirko Boland's cross was met on the volley by Reichel, who drilled the ball into the top corner from a tight angle, leaving Bernd Leno no chance and turning the BayArena silent.
Leverkusen responded with their best chance of the game so far, a Lars Bender effort which Daniel Davari pushed away from his goal.
In the 52nd minute, Leverkusen were gifted a route back into the game when Marcel Correia handled the ball inside the penalty area.
Kiessling stepped up to the spot to score for the fourth game running, taking his tally for the season to 14.
It only relieved the tension marginally, and only momentarily too as Leverkusen's failure to push for a second soon saw the fans turn against them.
Faced not only with a fighting Braunschweig side, the Leverkusen players also had to cope with the growing weight of expectation and, although they increased the pressure on their opponents late on, they could not create any clearcut chances.
In the very last minute, they almost lost the game too with Boland's shot saved well by Leno, although he could not save his side from the fans' frustration come the final whistle.