When Chris Harris was bowled by Chris Gayle in the 44th over of New Zealand's innings and his team stood on 188 for seven, the Kiwis seemed to be tumbling out of the World Cup.
Fleming sounded relief after the win in Port Elizabeth
But the all-round brilliance of Andre Adams turned things round in no uncertain finish to leave Group B looking very interesting indeed.
And the Black Caps captain put it succinctly. "We are back in the competition," said Stephen Fleming.
The man often regarded as the best captain in contemporary cricket added: "There was a lot of pressure on us today.
"It was a workmanlike performance which is symbolic of our team, our commitment.
"The Brian Lara run-out was crucial, it turned the match. We have a big game next against South Africa and look forward to it."
Chris Cairns was the man who effected the run out with a direct hit from around 40 yards.
And though he batted well, his bowling, the first time he had turned his arm over in a one-day match for a year, was less than brilliant as he conceded 21 runs in his only over.
Fleming said: "He was nervous and that's expected. He's bound to be a bit rusty but he will get better with more work."
Cairns himself joked: "I did OK with the bat. Flem brought me on to bowl to keep the Windies in the game. It was a bit disastrous but it's great to be back out there.
"I've got a lot of hard work to do but hopefully as we progress in the tournament I'll just get better. But the way the guys did out there today was just brilliant."
In a clear reference to how New Zealand's expected boycott of their match in Kenya will put them under huge pressure, Cairns added: "Every game for us in the tournament from now on is just desperation."