The Russians, who lost to the U.S. in last year's final, earned their decisive point in the 3-2 win over Serbia when Djokovic withdrew from the first reverse singles match despite leading Nikolay Davydenko 6-4, 6-3, 4-6.
"I decided to stop because I don't want to risk my health," the Australian Open champion said. "I wanted to fight. I tried to play, but in the end I was aware of the fact that I am very dizzy and I have a lack of energy. I just didn't want to continue because of the risk I had."
Sweden rallied to beat Israel 3-2, winning both reverse singles matches Sunday, and Philipp Kohlschreiber gave Germany the winning point in a 3-2 win over South Korea in the best-of-five series by beating Lee Hyung-taik 6-0, 4-6, 6-1, 7-6 (1).
Djokovic, who was playing his first singles match since winning the Australian Open, withdrew from the opening singles Friday because of the flu. But he played doubles Saturday, partnering with Nenad Zimonjic to give Serbia its first point.
Russia will play the Czech Republic in the quarterfinals in April. The Czechs lost both reverse singles matches Sunday but still beat Belgium 3-2.
The United States, which won its record 32nd Davis Cup title last year, took a 4-1 victory on clay at Austria. The U.S. will face France in the next round.
Stefan Koubek earned Austria's lone point when Mike Bryan retired while trailing 7-5, 1-0. Bryan's brother Bob, the other half of the winning doubles team, defeated Werner Eschauer 6-0, 3-6, 7-6 (3) in the final match.
The French team swept Romania 5-0 with Michael Llodra and Arnaud Clement winning Sunday.
Thomas Johansson and Jonas Bjorkman kept Sweden in the competition at Israel. Johansson defeated Dudi Sela 7-6, 6-1, 7-5 to even the score at 2-2 and Bjorkman recovered from a horrible first set to beat Harel Levy 0-6, 6-4, 6-3, 7-6 (6).
The Swedes, who reached the semifinals last year, will play Argentina in the quarterfinals.
The Argentines beat Britain 4-1, with Jose Acasuso defeating Alex Bogdanovic 7-5, 7-5 in the first reverse singles match and Jamie Baker giving the visitors a point by beating Agustin Calleri 7-6 (7), 6-4 in the second.
Kohlschreiber got his third win of the series to help Germany advance past South Korea, serving two aces in the fourth set tiebreaker.
"The whole week has been great for me," Kohlschreiber said. "I dominated from the start."
The Germans played without Tommy Haas, who has a shoulder injury.
Germany next gets Spain, which led 3-0 at Peru ahead of Sunday's reverse singles matches.