Hannah Whelan, Becky Downie, Rebecca Tunney, Claudia Fragapane and Ruby Harrold finished with a team total of 170.663 to come second behind Romania who won with 172.754. Russia finished third, more than a point behind Britain.
Downie scored an impressive 14.833 on bars and also top scored for Britain on vault (14.700) and beam (14.133). Fragapane claimed the best score of the day on floor with 14.766, pushing the experienced Vanessa Ferrari of Italy into second.
The silver medal matched Great Britain's best result at a European Championships, achieved in Birmingham four years ago and delighted British Gymnastics head of performance Roy Headey. "Silver is a great result for the team," he said. "We always have one eye on the 2016 Rio Olympics and today gave me a lot of belief about how we are progressing.
"Our task now is to analyse how Romania are achieving their scores and see how we can make gradual improvements so we can get up to that level ourselves. It's really pleasing to see how the team performed under pressure today."
Britain started their medal bid on the beam. Two-time European beam bronze medallist Whelan put in a strong performance to help the team to fourth after the first rotation. Fragapane, a 16-year-old debutant, then impressed on the floor, before Downie produced a stunning vault as Britain moved up the leaderboard.
Downie, Tunney and Harrold then kept their composure on the final apparatus, the bars, to move top of the leaderboard for a brief time, before Romania edged them out with their strong floor routines.