Nalbandian secured a 7-5, 6-3, 4-6, 6-4 victory to clinch the series at Stockholm's Royal Tennis Hall.
Nalbandian broke for a 2-1 lead in the fourth set and held his serve the rest of the way, converting his first match point with a backhand winner.
Story continues below ↓advertisement | your ad hereEarlier Sunday, Sweden's Robin Soderling beat Leonardo Mayer 7-5, 7-6 (5), 7-5 to level the tie.
Neither Vinciguerra nor Nalbandian was originally set to play the final match. Vinciguerra replaced Joachim Johansson and Nalbandian decided shortly before the start that he was fit enough to play instead of Eduardo Schwank.
Nalbandian missed much of last season with an injury, and said he made up his mind just 10 minutes before the match.
"I didn't feel very fit yet, but today it was enough," Nalbandian said. "I warmed up, and I felt worse than I thought. ... But with the flag behind us, it's tough to not be on court."
Vinciguerra took a 3-1 lead in the first set before Nalbandian found his stride. The Argentine then controlled play until the final game of the third set, where he saved three break points before netting a forehand on the fourth to let the Swede stay in the match.
"It was tougher than I thought," Nalbandian said. "The last point is never that easy in Davis Cup."
The seventh-ranked Soderling defeated Schwank in the opening singles Friday and Mayer beat Johansson to level the tie. Nalbandian and Horacio Zeballos then defeated Soderling and Robert Lindstedt in the doubles on Saturday.
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