The pre-match hype surrounded the return of Ben Stokes but his was a short-lived contribution while Buttler and Root held the innings together.
Stokes, returning for his first international since a nightclub incident five months ago, lasted 22 balls and was gone for 12.
Root, however, made 71 off 75 deliveries while Buttler belted 79 off 65 before he was run out off the penultimate ball.
England were looking at a score of more than 300 until tight New Zealand bowling restricted them to just 31 runs between the 42nd and 48th overs.
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Initially, he looked out of touch having not been required in the recent Twenty20 series, and he nicked Boult just wide of Martin Guptill at second slip.
But once settled into his work, the runs flowed for Root who brought up his 50 from 53 deliveries.
There were signs Root was starting to tire when he gave a chance to Mitchell Santner on 69 and then only added two more runs before he was bowled by New Zealand part-timer Colin Munro.
Buttler joined Root in the 29th overs when Stokes was dismissed with England 139 for four, and after facing seven scoreless balls he upped the tempo.
At the top of the England innings, Bairstow looked very much a batsman who had not played in recent weeks as he played and missed repeatedly outside off stump before eventually making a connection and edging the ball to Taylor to be gone for four.
His fellow opener Jason Roy, who struggled for runs in the recent Twenty20 tri-series, survived a caught and bowled chance to Sodhi on 39 and went on to make 49.