Russia entered Sunday's action with a 2-0 lead, but Carla Suarez Navarro brought Spain within one in the day's first reverse singles match with a 6-0, 6-3 straight-set victory over Nadia Petrova at Olympic Stadium.
Soler-Espinosa then battled back after dropping the first set to Kuznetsova, but failed to convert on her two break chances in the third to fall 6-2, 4-6, 6-3.
In a dead doubles rubber on Sunday, a Spanish team of Nuria Llagostera Vives and Arantxa Parra Santonja was leading a Russian tandem of Kuznetsova and Petrova 6-3 after one set when the Russians retired, as the hosts settled for a 3-2 victory this weekend.
The five-time champion Spaniards were playing in the Fed Cup's World Group for the first time since 2009. They fell to 1-6 against the Russians in Fed Cup play.