Captain Dean Couzins was yellow carded for a trip on England skipper Barry Middleton and Jackson scored from the resulting penalty corner.
Mark Gleghorne added the third from another penalty corner, when New Zealand goalkeeper Hamish McGregor might have done better, low to his right before New Zealand's only goal 19 minutes from the end.
Inglis was on hand to scramble the ball into the England goal after the Black Sticks' second penalty corner.
However Tom Carson and Middleton added more goals for England in the last four minutes to blow out the scoreline.
New Zealand defenders Alex Shaw and Blair Tarrant did good work at times when the goal was under pressure while Child, Inglis and Arun Panchia were prominent.
As well as Couzins sitting out five minutes, the Black Sticks lost Steve Edwards for five minutes late on for a second green card.
England had 15 shots on goal to New Zealand's eight, 22 circle penetrations to 18 by the Black Sticks and 54 percent of possession in the opponents' half.
New Zealand will have some hard thinking ahead tomorrow before Wednesday's quarter-finals. For starters they have managed only two goals from nine penalty corners in the tournament.
New Zealand must win the quarter-final if they are to achieve the goal of a top four finish.