Roger Schmidt's side twice went ahead but were forced to settle for a draw as they lost their 100% record in the Bundesliga after a late strike from Sebastian Prodl
Leverkusen came into the match as the only team to have won their first two fixtures.
However, after going ahead through Tin Jedvaj during a dominant first-half display, the Bundesliga's early leaders saw Bremen wrestle back the advantage through Fin Bartels and Franco Di Santo.
Hakan Calhanoglu and Son Heung-min hit back during the second half to put Leverkusen back in front, but Sebastian Prodl's late strike ensured a share of the spoils for the visitors.
Calhanoglu fired a thunderous effort against the crossbar 11 minutes in, before Jedvaj broke the deadlock six minutes later.
Stefan Kiessling found Jedvaj in acres of space, leaving the 18-year-old to smash home his second goal for the club and Leverkusen's 2,000th in the Bundesliga.
Kiesling should have doubled his side's advantage moments later, but his reaction shot from a Raphael Wolf parry was weak.
It proved to be a frustrating opening half for Leverkusen, despite the goal, as they hit the woodwork on two more occasions and failed to build on their early lead.
And they were punished on the stroke of half-time when Bartels pulled the trigger from 18 yards to find the bottom-right corner.
After all their chances in the first half, it looked like Leverkusen might come away empty-handed when Di Santo slipped the ball through the legs of goalkeeper Bernd Leno on the hour mark.
But the home side were back level three minutes later - Calhanoglu curling home a free-kick from just outside the penalty area.
The recovery appeared complete 17 minutes from time when Son turned brilliantly on the edge of the box to drill home a left-footed effort.
However, Prodl had the final say - smashing in from close range after a cross from the right had been flicked into his path by Nils Petersen.