She finished third in Thursday's qualification round with a best jump of 14.02m, but was outside the podium positions going into the sixth and last jump in the final.
Having jumped 13.64m in her second jump, she fouled three times in a row to lie fifth going into the last round.
But the 29-year-old saved her best jump of the day until last, recording a well-executed 13.99m to take her above Turkey's Tugba Danismaz and Lithuania's Dovile Kilty.
Danismaz had one final jump to try and knock Salminen off the podium, but could not improve on the 13.79m she jumped in the opening round.
A first major medal is redemption for Salminen, whose 2024 Olympic competition ended prematurely with injury in the qualification rounds.
The gold medal went to Spain's Ana Peleteiro-Compaore, who regained the title she last won in 2021 with a jump of 14.37m, clear of Diana Ana Maria Ion of Romania, who took silver, jumping 14.31m.
Fifth for Hurske in 60m hurdles
Earlier today, Reetta Hurske and Lotta Harala both made it through their Semi-finals to book two places for Finland in the Final of the women's 60m hurdles.
A season's best of 7.96 seconds meant Hurske was the fifth fastest in the Semis, while a personal best of 7.97 seconds allowed Harala to sneak into the Final as the eighth fastest.
There would be no such joy for Harala in the Final, however, as she clattered into the first barrier and fell, ending her race before it had really begun.
Hurske ran a fluid race, but was no match for the winner, Ditaji Kambundji of Switzerland, who set a new European record of 7.67.
The Netherlands' Nadine Visser was second in a new national record of 7.72, while Pia Skrzyszowska of Poland ran a season's best to take the bronze medal.
Hurske could not improve on her Semi-final time, finishing fifth in 8.00 seconds, behind Ireland's Sarah Lavin.
