Nalbandian, who is 153rd in the world, won 6-4, 7-6 (7/5), 7-6 (8/6) in two hours 46 minutes to record his seventh win over the 29-year-old Davydenko in their 12 head-to-head meetings.
"We both didn't play our best today. I didn't like the match as we both were making plenty of mistakes," Nalbandian said. "But I won and that's the most important thing about this tie.
"But I need to improve my playing rapidly to perform well in the remaining matches."
Davydenko started well, making a break in the third game of the opening set, but Nalbandian broke back immediately to level before producing another break in the 10th game to gain a 1-0 lead in 45 minutes.
Davydenko, currently the sixth in the ATP rankings, stepped up a gear in the second set breaking twice from the start for a 3-0 lead before Nalbandian replied with a couple of breaks to level at five games all.
In a tiebreak that followed Davydenko was leading 5-2, but Nalbandian picked up steam to win five consecutive points for a two-set advantage after one hour 48 minutes on court.
The rivals traded breaks throughout the third set but failed to decide the winner forcing another tiebreak. The 28-year-old Nalbandian peformed at more accuracy again winning the set and the match to put his team in front.
"It was very difficult for me to play the first match (in the tie) for Russia," Davydenko said. "I didn't play many matches recently and lacked confidence today.
"I played at 100 percent of my current form but it was not enough to win today."