The win at Dresden, Germany, left England in second place in Group B behind Japan, which earlier beat Mexico 4-0.
New Zealand struck first with a goal by Sarah Gregorius 16 minutes into the contest. It was the first time a New Zealand team ever led in a World Cup match.
Jill Scott tied it for England in the 63rd minute and Clarke, assisted by Scott, followed with the clincher high in the net shortly afterward.
England and Japan will meet Tuesday. Japan has already clinched a quarterfinal berth. England needs a win or draw against the Japanese to reach the quarters.
New Zealand lost its first two games and is out of contention.