The U.S. will face the Spain-Morocco winner in the second round.
Spain's Juan Balcells and Alex Corretja rallied past Morocco's Karim Alami and Hicham Arazi, 2-6, 6-2, 7-6 (10-8), 6-4, giving the Spaniards a 2-1 advantage.
U.S. captain Patrick McEnroe chose to leave hard-serving veteran Todd Martin on the bench. Instead, he paired the 21-year-old Blake and 20-year-old Fish and they responded to the challenge.
Despite dropping the second set, Blake and Fish were buoyed by the partisan crowd at the Myriad Center. They broke the Slovakian team's serve three times on the way to their win.
Blake was participating in just his second Davis Cup tie, while Fish was making his debut in the competition. But they still managed to halt the Americans' losing streak in Davis Cup doubles matches at five -- the longest in U.S. history.
In other action, the Netherlands stayed alive against defending champion France. The Dutch team of Sjeng Schalken and Paul Haarhuis defeated the French duo of Fabrice Santoro and Cedric Pioline, 6-4, 4-6, 6-1, 7-5 at Metz, France, reducing the Netherlands' deficit to 2-1.
Australia, the 2001 Davis Cup runner-up, lost in the opening round this year to Argentina.
Guillermo Canas and Lucas Arnold outlasted Australia's Wayne Arthurs and Todd Woodbridge, 3-6, 6-3, 6-4, 1-6, 10-8, to give the Argentines a 3-0 lead.
The Australians are playing without world No. 1 Lleyton Hewitt, who is recovering from the chicken pox, and former U.S. Open champion Pat Rafter, who is taking time off from the game. Mark Philippoussis, who struggled with a knee injury in 2001, still is trying to regain full fitness.
The Czech Republic also advanced to the second round on Saturday. The Czech tandem of Jiri Novak and David Rikl posted a 6-4, 6-3, 7-5 win over the Brazilian pair of Alexandre Simoni and Andre Sa at Ostrava, Czech Republic.
Croatia took a 2-1 lead over Germany when Goran Ivanisevic and Ivan Ljubicic overpowered Michael Kohlmann and David Prinosil, 6-4, 6-4, 7-5 at Zagreb, Croatia.
Great Britain and Russia also opened 2-1 advantages at home.