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GB finished bottom of the Pool stage in London with no wins and only three goals scored, but produced their best performance in an enthralling fifth/sixth place play-off.
Alex Danson scored a match-winning hat-trick, with Lily Owsley also on target.
"We didn't lose belief and we are delighted," Danson told BBC Sport.
"Obviously I'm delighted [with the three goals] but not just for me, from a whole team perspective because we can build on that performance leading into the summer."
Head coach Danny Kerry admitted "harsh words" were spoken after Britain's 4-1 defeat by Australia on Saturday, but was please with how his team responded.
"We've had a tough tournament and they know that yesterday wasn't good enough, but today they really committed," he told BBC Sport.
Danson struck her first goal of the tournament in the 10th minute, but Anita McLaren drew New Zealand level in the second period.
Owsley and Danson struck either side of half-time to take GB clear, but back came the Kiwis with goals from Olivia Merry and Kelsey Smith made the score 3-3.
Danson needed time out after being struck in the head by the ball, but recovered to deliver a composed lofted finish over New Zealand keeper Sally Rutherford to win Britain the match.
"Although we conceded a few today, which we need to work on, overall that was much better," captain Kate Richardson-Walsh told BBC Sport.
"We have a lot to learn, huge amounts, but it's good we're doing that now and can take that forward to Rio."
British selectors will reveal the 16 women and 16 men who have been chosen to represent Team GB at this summer's Rio Olympics on Tuesday.
Britain's women beat New Zealand 4-3 to secure a first win at the 2016 Champions Trophy, in what was their final match before the Rio Olympics.