Downie, 15, became the first British woman to win an international all-around medal in her first senior year.
Purvis also finished third in the men's competition to claim his second career European all-around bronze.
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Fellow Britons Sam Oldham and Claudia Fragapane finished sixth in their respective overall competitions.
Downie, in her first senior year, scored 56.623, behind winner Giulia Steingruber of Switzerland and Russia's Maria Kharenkova.
"I'm over the moon. I never thought coming here I'd be among the medals," Downie told BBC Sport in Montpellier. "I was hoping for a top six or eight finish - it's just absolutely amazing."
Downie, younger sister of European uneven bars champion Becky, made a good start on bars then followed with solid routines on beam and floor - where she was marked down for stepping out of the field.
She finished strongly with 14.8 on the vault - her strongest apparatus, and the one where she will aim for another medal in Saturday's final, as well as the uneven bars.
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"I was really nervous going into the vault," she said.
"I knew if I nailed it I could have a shot at a medal so I just tried to keep calm and do what I do in training."
Britain's second medal of the day came thanks to 24-year-old Purvis who scored 87.423 over his six apparatus routines.
Ukrainie's Oleg Verniaiev took the all-around gold ahead of David Belyavskiy of Russia.
Ellie Downie and Dan Purvis each delivered Great Britain a bronze medal at the European Gymnastics Championships in Montpellier.