Sri Lanka and the West Indies eyed up semi-final places after hard-fought wins in group one of the Super Eights, with the hosts needing a dramatic one over eliminator for victory.
With the scores level at 174, Sri Lanka needed one run off paceman Tim Southee's last ball but Lahiru Thirimanne failed to reach the crease after attempting a sharp single.
New Zealand skipper Ross Taylor had deflected the ball from his knee to the stumps after fumbling the throw, which took television umpire Steve Davis of Australia several minutes to rule the batsman out.
Chasing 175 to win, Sri Lanka were well on course for a comfortable win but Tillakaratne Dilshan's run out after scoring a brilliant 76 helped New Zealand tie the regular match.
Sri Lankan skipper Mahela Jayawardene admitted that the tense finale had been "chaotic".
"We were looking at Ross's reaction. He wasn't that enthusiastic about the run out so we thought we had scraped through. It was chaos down there," said Jayawardene.
"To be in a situation where you think you've won and then go back and play was tough but we kept our composure and made sure that we played some smart cricket," said Jayawardene, whose team scored 13 to New Zealand's eight in the Super Over.
Jayawardene, whose team next faces the West Indies here on Saturday, hopes to carry on the momentum from the victory over the Black Caps.
"When you are leading a team in a pressure situation and a crunch game like this, points on the board are very important. We can surely build on this," he said.
Sri Lanka have not won a major trophy since their 1996 World Cup win in the sub-continent. They were runners-up in the 2007 and 2011 World Cup (50 overs) and lost to Pakistan in the 2009 World Twenty20.
West Indies, who gained two crucial points by outsmarting title-holders England by 15 runs, are also looking for a first major trophy in eight years.
"When we left the Caribbean, winning this tournament for our fans was on our minds. The last decade has been tough for us. We last won silverware in 2004," said Sammy of West Indies Champions Trophy win in England.
"It is a good opportunity for us to do that here. We've been playing well as a team and we have to take it one game at a time," said Sammy, who rated Sri Lanka as one of the big teams.
"The senior players always turn up for Sri Lanka and they have a good mix on the squad. You just can't count them out at home. Playing them is a big challenge but it's one we're ready for," said Sammy.
West Indies rode on a brilliant 103-run opening stand between Johnson Charles (84) and Chris Gayle (58) to post 179-5 before surviving a late onslaught by Eoin Morgan (71 not out) and Alex Hales (68).
Sammy said fast bowler Fidel Edwards will come under consideration after leg-spinner Samuel Badree was brought to tame England with slow bowlers.
Sri Lankan bowler Lasith Malinga celebrates with teammates after taking the wicket of New Zealand's Martin Guptill in the super over during the Twenty20 Cricket World Cup's Super Eight. Sri Lanka and the West Indies were eyeing semi-final places after hard-fought wins in group one of the Super Eights.