Nelsen ended his seven-year spell at Blackburn on deadline day when he had his contract cancelled and moved immediately to Spurs on a free transfer.
During his time at Ewood Park, Nelsen reached two FA Cup finals - one in 2005 when Rovers lost 3-0 to Arsenal, and another two years later when they were defeated 2-1 by Chelsea after extra time.
Now the New Zealander, who is likely to feature in Tottenham's fifth-round game at Stevenage on Sunday, is desperate to go one better with his new club, who he thinks have a good chance of going all the way this year.
"I've been to three semi-finals, two of them in the FA Cup, so it would obviously be nice to push on," said Nelsen, who also experienced semi-final heartbreak in the Carling Cup in 2010.
"I've had a bit of a taste of it and have come so close to getting to a final so I feel I've got a bit of unfinished business with the competition."
Tottenham, third in the Barclays Premier League, are favourites to clinch their first FA Cup in 21 years after both Manchester sides were eliminated from the competition.
Nelsen has warned his new team-mates they must not start thinking a place in the final is guaranteed, however, and knows that npower League One side Stevenage, who are 47 places below them in the league ladder, will be determined to cause an upset at Broadhall Way this weekend.
"This squad is good enough to go on and win it, and I know it's a priority here," Nelsen said. "But we'll be taking this game seriously because it's going to be a very awkward one.
"It's easy to look at it like we are favourites (to win the competition) on paper but the FA Cup is never like that.
"There's always banana skins. They're very difficult games and if you don't get the mental side of it right, it can come back and haunt you, hurt you.
"We've all witnessed it in the past so we'll just take it one game at a time. Hopefully we'll be all right for Stevenage. We definitely won't take it lightly.
"They beat Newcastle (two years ago in the third round). They'll be a hard team to beat."
Nelsen made his Spurs debut as a second-half substitute in Saturday's 5-0 drubbing of Newcastle at White Hart Lane.
The 34-year-old had only previously made one appearance for Blackburn this season and had spent most of his time in the Lancashire club's reserves, making last weekend's experience all the more sweeter for the veteran centre-back.
"It was just an absolute honour to get my debut for Tottenham," he said.
"White Hart Lane is my favourite stadium. I can remember playing here five or six times, obviously normally on the opposing side, so when you come out [for Spurs] it's a whole different feel.
"It's an amazing stadium and fantastic fans, so it was great to get my debut.
"It couldn't have gone better. Newcastle are obviously having a great season, so to get three points was brilliant."
With Ledley King likely to be rested, and Younes Kaboul and William Gallas both struggling with injuries, Nelsen may get his first start for Tottenham on Sunday afternoon.
He accepts first-team opportunities will be hard to come by when Harry Redknapp's first-choice defenders are fit, but he is content with mucking in when needed.
"When I was asked to come here it was just to help out in any way I could, on and off the field," Nelsen added.
"They've just got some incredible players here.
"It's great to have squad strength like that, and we've got it in every single position. It's the best squad in the league."