Spence, who won World Championship gold in Rome last year, had managed just 33rd place in the previous World Cup outing this season in Rio but this time around she found her form in the Hungarian capital.
After placing sixth after the fencing event with 904 points, Spence moved up to fourth after the swim, racking up 1208 points and after a flawless ride, she found herself in joint first place going into the combined run/shoot event.
Ukraine's Ganna Buriak and Spence began the final discipline 11 seconds in front of the chasing pack and while Buriak held on for the win and her compatriot Victoria Tereshuk moved into second place, the Brit held on for bronze, 28 seconds behind the winner.
Katy Burke, who began the final round in seventh place after two good outings in the fencing hall and in the pool, slipped down to 22nd.
Double Olympian Sam Weale and fellow Brit James Cooke will contest the men's final on Sunday.
- Sports & Recreation