The scores of the nearly five-hour clinching match were 6-3, 6-4, 5-7, 3-6, 17-15.
Russia, which took a 2-0 lead Friday, had not beaten the United States in two previous Davis Cup ties, losing in the 1995 final and dropping a first-round battle in 1998.
"I was in control from the beginning, but suddenly lost my nerve," Tursunov said. "It was very hard to come back in the deciding set but I knew I could do it."
At 11-12, Roddick saved three match points on serve to stave off defeat, but Tursunov converted his fourth match point opportunity nine games later.
Russia will play Argentina for the Davis Cup crown in December. Argentina Saturday clinched its semifinal against Australia.