Nalbandian fell behind 4-0 and failed to mount much of a challenge in the first set. But the 30-year-old Argentine veteran dominated the next two sets. He fired an ace to serve out the third set before Mayer began playing well again.
The German broke for 2-0, saved three break points to lead 3-0 and held serve for 4-1. But Nalbandian finally broke back in the seventh game.
A net-cord helped Mayer take a 6-5 lead, but Nalbandian held to reach a tiebreaker. He had five match points when leading 6-1 but allowed Mayer to pull back to 6-5 before converting his fifth.
Canada 1, France 1
Milos Raonic beat Julien Benneteau 6-2, 6-4, 7-5 on Friday night to give Canada a 1-1 tie with France after the opening singles in the Davis Cup match, Canada's first at the World Group level in eight years.
The winner of the best-of-five match will play the U.S.-Switzerland winner in the quarterfinals. The visiting United States leads that match 2-0.
Spain 2, Kazakhstan 0
Nadal and David Ferrer have helped Spain win three of the past four titles. Both are absent, forcing new captain Alex Corretja to overhaul the squad.
Ferrero, a former No. 1, and his 11th-ranked teammate put Spain on course for a 22nd straight victory at home. On Saturday in the best-of-five series, Spaniards Marcel Granollers and Marc Lopez play Evgeny Korolev and Yuriy Schukin in doubles.
Czech Republic 2, Italy 0
The Czechs need one more victory to win the best-of-five series in the World Group. They are expected to use Berdych and Stepanek in Saturday's doubles against Daniele Braccialli and Potito Starace. Reverse singles are Sunday.
The winner will face Serbia or Sweden in the quarterfinals in April.
The ninth-ranked Tipsarevic had 11 aces and converted four of five break points to beat Prpic 6-3, 6-3, 6-4 in the best-of-five World Group series.
Prpic failed to take any of his five break chances. Prpic is ranked No. 1,426, but Tipsarevic says "the rankings don't do him justice."
Troicki downed Ryderstedt 6-4, 6-3, 5-7, 6-3 after missing his chance to finish the match in the third set, when he broke serve for a 4-2 lead.
Serbia, which beat Sweden 4-1 last season, is playing without top-ranked Novak Djokovic, who is resting after winning the Australian Open.
Austria 2, Russia 0
Austria needs one more win in the best-of-five series to enter the quarterfinals of the World Group for the first time since 1995.
Russia has managed to win from 0-2 only once, against Germany in 1995.
The 34th-ranked Bogomolov, who has been living in the United States since 1992 but opted to play for his country of birth, looked nervous on his Davis Cup debut. Bogomolov dropped serve three times in the fourth set and lost the match on his third double fault.
Croatia 1, Japan 1
Nishikori became the first Japanese man to reach the quarterfinals of the Australian Open since the Open era began in 1968, but struggled against the big-serving Croatian.
Japan is in the World Group for the first time in 26 years.
Information from The Associated Press was used in this report.