Four falls over beam and uneven bars meant the British team finished nearly five marks off the gold spot but still nearly two ahead of France in third.
"We'd be lying if we said we weren't disappointed but I'm still incredibly proud," Becky Downie told BBC Sport.
"To have that many falls and still get silver shows how much we've improved."
She added: "Two years ago we were crying with happiness because we scraped a silver."
The British team of Becky and Ellie Downie, Claudia Fragapane, Gabby Jupp and Ruby Harrold were within touching distance of Russia after the first apparatus, with the younger Downie sister Ellie posting a fine 15.1 on bars.
Becky Downie scored 15.066 after Harrold had missed a catch in her routine.
The elder sister then suffered falls on beam, as did Fragapane, while Russia, led by Aliya Mustafina, extended their lead.
However, Fragapane was able to put her obvious disappointment behind her to give trademark energetic performance on the floor to seal a comfortable silver medal with a total score of 170.312. Russia won the gold medal with 175.212, and France were third on 168.496.
The GB team in Switzerland travelled without Amy Tinkler, who remained at home to concentrate on school exams, although they did welcome back Jupp after three years plagued by injuries.
On Sunday, Britain will have seven chances to win medals across all four apparatus.
Great Britain's women won team silver behind Russia in the European Championships despite not being at their best in the final in Bern.