Youzhny beat Philipp Petzschner, 6-4, 6-4, 3-6, 6-3, while Andreev trounced Philipp Kohlschreiber, 6-3, 3-6, 6-0, 6-3.
"Every match in Davis Cup for your country is very important for everybody, it doesn't matter what the score is," said Youzhny. "If you come to play for your country, you have to be ready. That's why it's an important match for me and an important match for our country."
The defending champion Russians began the day trailing the best-of-five tie, 2-1, following a doubles loss Saturday.
The victory means they will defend their title later this year against the United States, a 4-1 winner over Sweden.
The Americans will be the host nation for the final Nov. 30-Dec. 2, when Russia will try for its second straight Davis Cup crown and third overall.
Swim Week Miami Beach 2014 [PHOTOS]