Trailing 2-1 from the first leg, Sweden needed an upturn in recent form to win bronze in front of their home crowd at Strawberry Arena in Solna.
But not only did the first 45 minutes pass without them registering a shot on goal, their plan went awry just before the hour mark when Clara Mateo put France ahead on the night and 3-1 up on aggregate.
However, Blagult kicked into gear in the final stages - six minutes from time, Evelyn Ijeh made it 1-1 on the night with her second international goal, before Brighton's Rusul Kafaji popped up in the 92nd minute to make it 2-1 and send the tie to extra time.
Sweden made five changes in the 90 minutes in an attempt to change the game, meaning they only had one substitute available in the extra period, compared to France's four.
Those changes proved crucial, as Gago was thrown on six minutes into extra time, and 10 minutes of play later, she converted what turned out to be the winner, as it finished 2-2 on the night, with Les Bleues taking it 4-3 on aggregate.
It's now five games without a win for Sweden, stretching back to their elimination at the Quarter-final stage of UEFA Women's EURO 2025.