Dortmund looked on course to maintain their perfect start after Daniel Svensson and Nico Schlotterbeck scored in quick succession during the second half. However, Ludovit Reis pulled one back late on before Fullkrug headed home in stoppage time to complete a remarkable comeback against his former club.
The hosts dominated possession during a largely quiet first half, with Marcel Sabitzer firing over and later striking the stanchion with a free-kick. Maximilian Beier also went close, but Bremen goalkeeper Karl Hein produced an excellent one-handed save to keep the scores level.
Dortmund increased the pressure after the break, with Hein denying efforts from Schlotterbeck and Arsenal loanee Ethan Nwaneri. Fullkrug nearly punished his former employers at the other end, but goalkeeper Gregor Kobel kept out his close-range attempt.
Nwaneri thought he had a chance to break the deadlock with a spectacular overhead kick, only for the effort to be ruled out for an infringement in the penalty area.
Dortmund eventually made their pressure count on the counter-attack. Serhou Guirassy drove forward down the right and delivered the ball into the box, where Fabio Silva's scuffed effort fell kindly for Svensson to slide home and mark his 50th Bundesliga appearance with a goal.
Schlotterbeck then doubled the lead five minutes later, heading home Joey Veerman's corner to leave Bremen facing defeat.
However, the visitors refused to give up. Schlotterbeck's misplaced pass allowed Reis to seize possession and fire past Kobel in the 88th minute, setting up a tense finish.
Bremen completed their comeback when Marco Grull delivered a floated ball into the box and Fullkrug powered a header into the net, extending his scoring run to four consecutive Bundesliga matches.
The former Dortmund striker was delighted with his team's resilience and said the atmosphere reminded him of his European nights with BVB.
"It felt a little bit like a Champions League night, like the ones I experienced with BVB in the past," Fullkrug said. "We showed so much commitment in our performance tonight."
He explained that conceding twice had forced Bremen to adopt a more attacking approach.
"After conceding twice, we had to open up a bit and push further forward. That gave us a new lease of life, which was incredible. Full credit to the team for that performance."
Fullkrug, who is on loan from West Ham, also played down the personal significance of scoring against his former club.
"The fact that I scored against BVB doesn't matter to me. What matters is that we came away with a point," he said, adding that the equaliser came from a set-piece routine the team had practised.
The result leaves Dortmund frustrated after surrendering a two-goal advantage, while Bremen can take considerable confidence from another impressive fightback at the Westfalenstadion.
Werder will next face newly promoted Paderborn before Fullkrug returns to Dortmund later in the month when the sides meet again in the DFB-Pokal.