Vinciguerra, the world number 34, left the courts without talking, but was clearly frustrated at his own performance.
At one stage he slammed his racket into the ground in frustration and was given a code violation warning for smashing a ball high over the grandstands.
Nielsen's serve was the difference between the two. He held his own serve throughout both sets, and in the first set tiebreaker kept an advantage from the first ball.
A small leap in the air was the only sign of delight when Nielsen won the first set, and in the second he enjoyed a break point in the third game before getting the vital service break in the fifth game.
Ahead 5-3, Nielsen had five match points on Vinciguerra's serve - all lost - before he served out for the match, converting his seventh match point by setting up a winning volley with a strong second serve.
Nielsen just shook his head about the five match points after the game, and said he tried hard to make Vinciguerra earn the game.
"I definitely wanted to finish it off there, I felt like I could have, especially with three match points in a row," Nielsen said.
"At the change of ends, I sat down and I knew I that I'd been serving well, and I knew that if I just worked that last game I could finish him off."
Nielsen said while Vinciguerra may have under-estimated him, that was not wise on the men's tour.
"There's always a good chance that with him ranked quite a lot higher than me that he wouldn't have thought too much about me," he said.
"It's a one-off match, it doesn't matter what your ranking is, the men's tour is so strong you can get through."
Two years ago Nielsen reached his best ever single's ranking of 172, but a poor 2001 season saw him slip to 457.
He has had four wildcard entries to the tournament before, and beat Brazilian Fernando Meligeni, then a top 60 player, in the first round in 1998.
But he said his win over Vinciguerra was bigger and sweeter, particularly with questions over whether he should get a wildcard entry this year.
"This year maybe I felt a little bit more pressure on myself to win. It felt like I didn't hit the ball that great from the back but I just had a lot of composure throughout the match," Nielsen said.
"I served well at the right times, also played some good points at the right time. I felt pretty calm throughout the whole match."
Neilsen will play the winner of the match between Michel Kratochvil of Switzerland and qualifier Robby Ginepri of the United States in his second round match on Wednesday. He will play in the first round of the doubles with James Shortall tomorrow night.
Vicente was a comfortable winner over Squillari, despite not feeling well after a marathon 20-hour journey here from Qatar where he reached the quarterfinals last week.
"It was tough to play because I was so tired, it's a long way. I'm a bit sick. Anyway, I win," Vicente said.
"I didn't do a lot of mistakes, and I was more consistent than him. No break points for him, I served well."
- NZPA
Heineken Open full coverage
Results
Tournament information
Tournament schedule
Players list and profiles
Ticket information
TV coverage
Related sites
ASB Bank Classic full coverage
Players list and profiles
Singles draw
Ticket information
Anna Kournikova picture gallery
ASB Bank Classic picture gallery
Related sites
From the sidelines: other news related to the tournaments
