"It's never nice to be on the wrong side of those tight ones."
Southee said New Zealand would have won but for a flustered middle-order display from several batsmen, who failed to give the Williamson the right support as he got within one big hit of winning the match off the last ball.
Henry Nicholls, Tom Latham, Mark Chapman and Colin de Grandhomme were removed for a combined 11 runs, undone by classy spin bowling from man of the match Moeen Ali and Adil Rashid, who shared five wickets.
Southee insisted the likes of Nicholls and Latham were more than capable but just needed confidence.
New Zealand will train today on the ground where they won their last match by 183 runs against Pakistan in January.
"We were obviously in the middle of a great run," Southee said. ''(But) the last wee while hasn't gone according to plan.
"We've come up against great opposition and we're in for a cracking last two matches of the series."
Meanwhile Ali has put a disappointing Ashes series behind him to re-stake his position as an integral part of the England side.
In 70 ODI overs against Australia and New Zealand, he has 10 wickets at an economy rate of 4.74. He is yet to go at worse than a run a ball in any of those eight matches.
The switch in the formats had "energised" him after a difficult period. "Once they [the Ashes] were done I'd pretty much got over them to be honest," he said. "You've just got to try and move on.''
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