Europe had built a two-point lead after the morning foursomes on Saturday and looked strong favourites to retain that advantage as the final quartet approached the closing holes.
But Kupcho had other ideas, draining a birdie putt from 91 feet on the 17th to bring her and teammate Lindy Duncan level against Carlota Ciganda and Celine Boutier going to the 18th.
Ciganda had been the hero of the opening day for Europe when she emerged from the rough to somehow secure a half on the final hole and ensure the two teams would be locked at 4-4 overnight.
But she failed to repeat the feat on Saturday, missing a birdie putt to keep Europe's hard-earned lead intact, before Kupcho produced another mighty putt from the edge of the green to snatch an improbable victory for the US pair.
Europe had made a big move in the morning session, with Maja Stark and Linn Grant teaming up to beat Allisen Corpuz and Nelly Korda 3&2 before Leona Maguire and Esther Henseleit won two of the final three holes to beat Kupcho and Rose Zhang.
A third European point came from Mimi Rhodes and Nastasia Nadaud, who dominated Andrea Lee and Auston Kim 4&3, but the woes of Europe's high-profile pair Charley Hull and Lottie Woad continued as they crashed 6&5 to Alison Lee and Lauren Coughlin.
Questions may be asked about European captain Anna Nordqvist's persistence with Hull and Woad as a partnership after they also fell in the fourballs, losing 1up to Yealimi Noh and Andrea Lee.
Zhang and Auston Kim beat Nadaud and Leona Maguire and what turned out to be Europe's only point of the afternoon session was carved out by Stark and Grant, who beat Americans Megan Khang and Coughlin 2&1.
Attention swung to the increasingly dramatic finale being played in the remaining match and Kupcho's pair of monster putts that sent her teammates dancing across the green in celebration.
