Britain’s previous medal in the event came in 2010, external when the squad – which included the now-retired Beth Tweddle – finished second in Birmingham.
Downie, who competed at the 2008 Beijing Olympics, is the only member of the 2010 line-up still in the GB team.
“It’s a totally different team [to 2010] with totally different expectations, and I think everybody really stepped up to the mark today,” the 22-year-old told BBC Sport.
“We really couldn’t have done any more, I’m really proud of everyone.”
Led by two-time Olympian Whelan, 21, Britain had led the qualification standings on Thursday and began with a solid display in the final on the beam which put them fourth in the rankings.
Despite an impressive routine from 16-year-old debutant Fragapane on the floor they remained in fourth place at the halfway stage.
They began to make progress in the third rotation as Beijing Olympian Downie produced a stunning vault.
Downie, Tunney and Harrold then delivered under pressure in the final event, the uneven bars, to secure Britain their place on the podium.