Tursunov prevailed 4-6, 5-7, 6-3, 6-4, 6-2 in a match lasting nearly four hours to give Russia a spot in the quarterfinals.
Buoyed by a surprise upset in the doubles Saturday, Hanescu converted his only break point to take the first set and had another two breaks in the second as Tursunov made 47 unforced errors in the first two sets.
But the Russian, who replaced Marat Safin, bounced back in the third. After trading breaks to make it 2-2, Tursunov broke again for a 5-3 lead and served out the set at love.
Tursunov only dropped three points on his serve in taking the fourth set and broke Hanescu in the first game of the fifth.
Gabashvili later beat Crivoi 6-4, 6-2 in a match lasting just more than an hour.
Though Crivoi had 10 aces, he also had six double faults to Gabashvili's one.
"It was really difficult to concentrate,'' Crivoi said. "I was also trying hard not to injure myself.''
Russia won the Davis Cup in 2006, then lost the final to the United States the next year. Romania has never won the tournament, and played its first Davis Cup match under new captain Andrei Pavel.
