Sangakkara started the day on 13,693 runs and needing just 12 more to pass Australia's Ricky Ponting who made 13,704 runs before he retired.
The Sri Lankan left-hander reached the milestone when he flicked Adam Milne down the leg side for a four in the 16th over of his team's reply to New Zealand's imposing 331 for six.
India legend Sachin Tendulkar tops the list for most ODI runs with a mammoth 18,426.