Twice champions Wolfsburg will meet the 2021 winners in the final in Eindhoven on Jun 3 after the Spanish champions eliminated Chelsea to reach the showpiece match for a third straight year.
With nothing to separate the two sides deep into extra time, the German team silenced the home crowd in the 119th minute when their high-pressing yielded an error for the goal.
Jule Brand pinched the ball from a tired Arsenal defence and laid it on a plate for Bremer to tap in from close range as Wolfsburg booked a spot in their sixth European final after losing the last three in 2016, 2018 and 2020.
"I feel empty at the moment. It was a very tough game and it's tough to concede a goal like that, but I'm so proud of how the team has done in the Champions League," Arsenal's Stina Blackstenius told DAZN.
"We've come back in so many games and gone so close to the final. I'm so proud but it's so tough to take."
An Arsenal side depleted by injuries fought back from 2-0 down to level the tie in the first leg in Wolfsburg but, roared on by a crowd of 60,063, it was the home side that took the lead in the 11th minute through Blackstenius.
Arsenal won the ball in their own half to launch a swift counter-attack and Blackstenius was put through by Lia Walti, with the Swedish forward jostling for the ball and capitalising on a defensive mix-up to put the ball into the net.
But Wolfsburg levelled from a free kick when skipper Alexandra Popp, returning from injury, headed the ball into the path of Jill Roord and goalkeeper Manuela Zinsberger saw the shot from the former Arsenal midfielder too late to make a save.
VAR RESCUES WOLFSBURG
Blackstenius scored again barely two minutes into the second half but VAR came to Wolfsburg's rescue to rule the goal out for offside and 10 minutes later the German side scored from a corner when Popp popped up at the near post with a header.
Arsenal levelled the tie again when Lotte Wubben-Moy retrieved a Wolfsburg clearance and crossed into the box for fellow defender Jen Beattie to direct a header past the keeper to force extra time where Wolfsburg's stamina shone through.
"I'm overwhelmed, it was a tough game with extra time. I thought it was going to go to penalties but we won," Roord said.
"Credit to Arsenal, they've been through a lot as a team but they made it super hard. At the end they could have won, we could have won. We were the lucky ones ... I'm looking forward to playing in the final in my own country."
Barcelona, who won the league this season with a perfect record, also reached the European final last year where they lost to Olympique Lyonnais.
