The 15-year-old Gutsu was third from last to go in the vault. She had watched as Miller, who had beaten world champion Kim Zmeskal in the US trials, pulled off two superb vaults, the first of which scored 9.975.
That left Gutsu needing 9.938 to take the lead, and she surpassed that mark on her second attempt.
Even then, Gutsu was not home and dry as her team-mate Svetlana Boginskaya, lying third after three of the four apparatus rounds, needed 9.965 from her beam routine to take over the gold medal position.
A slight wobble, however, meant that Boginskaya, who had led the former Soviet gymnasts to the team title on Tuesday, could manage only 9.912, a mark which dropped her down to fifth overall.
Miller won the silver medal but the biggest surprise of the competition was the failure of Zmeskal. A terrible mistake during her floor exercise cost her any chance of a medal and she could finish only 10th.
Romania's Lavinia Milosovici narrowly edged out her team- mate Cristina Bontas for the bronze.
Vitali Shcherbo, of the Unified Team, who topped the men's individual standings after the team compulsory and free exercises, is firmly established as favourite for tonight's all-round competition.
Shcherbo, second all-round at the Indianapolis world championships last year, is also in a strong position for gold medals in Sunday's apparatus finals, as he top- scored in four of the six disciplines in the team competition. 'I've been working for this for a long, long time and I hope all the hard work is going to pay off,' he said.