Whelan, 19, the 2010 and 2011 senior British champion, was successful on the beam and in the floor routine.
Her bronze on the beam was Great Britain's first ever medal in this discipline in a major championship.
Two 15-year-olds - Ruby Harrold and Rebecca Tunney - made their senior debuts, Tunney finished fifth on the floor, with Harold seventh on the bars.
The performance was a significant step for both Whelan and Team GB ahead of the Olympic Games.
Whelan was the senior member of the squad in the absence of Beth Tweddle and has never been on the podium before at this level.
"It feels amazing," said Whelan. "I'm over the moon, it's really unexpected.
"I wanted to come out and enjoy the competition and I definitely did. I enjoy training beam and feel naturally comfortable on it and to get our first-ever medal on that piece is very special.
"I've always dreamed of getting major medals and to come away with two is unbelievable. It gives me a lot of confidence going into the rest of the year."
Whelan's success followed a disappointing day on Saturday when Great Britain could only finish fourth in the team competition.
Romania were the team to beat in this year's tournament as they took home the team gold, and added three more individual golds on Sunday - Catalina Ponor on the beam, Larissa Andreea Iordache on the floor, and Sandra Raluca Izbasa in the vault exercise.
Great Britain's Gabby Jupp won bronze in the floor final of the Junior European Championships at the same venue.
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