The Kiwis stunned South Africa by 49 runs in the World Cup quarterfinals last night and Vettori attributed it to the team`s all-round performance.
"Every single bowler stepped up today, and probably it was the fielding that got us through. Martin Guptill was exceptional," Vettori said after the match.
Even as Vettori basked in his team`s glory, his rival Smith was a shattered man.
"It`s difficult to explain. There are no words to describe how I feel. Just have to take it on the chin. It`s been a very disappointing evening," Smith said.
Vettori said 250 would have been the ideal score on the wicket at the Shere Bangla Stadium here.
"Once we got 220, we knew we had to be aggressive. We were hanging in there even when Kallis and AB were going. We wanted 250 on this pitch. We lost some wickets, and scrapped through to 220.
"We knew it was a tricky wicket, and knew it was going to be an old-school one-day game. That`s the sort of game we like to play. Once we got de Villiers and Kallis, we were back in it. The run out of de Villiers was an amazing piece of work.
We are pretty happy to be in the semis, and will see how the wicket in Colombo plays tomorrow."
New Zealand have played in three semifinals since the tournament`s inception in 1975, and Vettori hoped the team will better that record.
"New Zealand have played in three semifinals, and hopefully, we can take two steps further this time around," Vettori said.
Smith was gracious in defeat and praised the Kiwis for their showing but was disappointed that his team lost too many wickets after sitting pretty at one stage.
"I thought we bowled pretty well. Credit to Jesse Ryder.
At 100 for 2, we were sitting well, and then we lost too many wickets. The stand between Jesse and Taylor kept them in good stead, but we kept them to 220, which is very gettable. We lost too many wickets."
"Your guess is as good as mine (referring to South Africa`s penchant for losing in knockout stages) its been happening since 1992," he said.
Despite the shock exit, Smith said there have been a few positives from the tournament.
"We have got some big positives out of the World Cup. We weren`t good enough tonight, but we have to take it forward and not get bogged down. There will be a new coach and new captain. We are disappointed with tonight, and nothing I say is going to change that for fans back home," Smith said.
Man of the Match Jacob Oram, who took four wickets besides taking a brilliant catch to dismiss Jacques Kallis, felt the South Africa all-rounder`s dismissal turned the match around.
"Thankfully I was tall enough and managed to get there for the Kallis catch. I don`t want to take credit for that one, but I suppose Kallis` dismissal was the turning point of the match," Oram said.
"It`s so hard when you are injured continuously to get any rhythm, so the feeling has been better to be on the field.
We believe we can go all the way now. This is among the top 3 moments in my career, another one being the win over South Africa in the 2003 World Cup in Wanderers."
"Considering that we were written off and doing well in a ground where we don`t have a great record, it`s a great feeling."