Astle, who was recalled to replace the injured Mitchell Santner, had Craig Overton trapped in front before he ended the test when Jimmy Anderson was caught by Trent Boult at mid-off to give New Zealand just their 10th test victory over England.
Paceman Neil Wagner's short-pitched tactics proved effective again as he dismissed both Ben Stokes and Chris Woakes, who had each scored defiant half-centuries and threatened to save the match for the visitors.
Wagner finished with 3-77, while Boult, who took 6-32 in the first innings, took 3-67. Astle finished with 3-39.
"It was a fantastic performance from us. If we go back to that first innings, we bowled great lengths and also got lucky to find the edges," New Zealand captain Kane Williamson said.
"They fought really hard but it was great we were able to fight the pressure and come through."
New Zealand would have felt aggrieved if the match ended in a draw after they destroyed England in the first innings, bowling them out for a record low of 58 against the Black Caps inside 90 minutes on the first day.
Persistent rain, however, washed out virtually two days of play, with England bowling just 23.1 overs on the second day and 17 deliveries on the third.
The final day had been eerily similar to a match from five years ago when New Zealand were well placed to win the third test against England at the same ground.
The tourists entered the final day on 90-4 facing an improbable task of scoring another 391 runs in 90 overs to win the match.
Ian Bell, however, scored 75 while wicketkeeper Matt Prior produced a rearguard 110 not out and protected last man Monty Panesar to ensure the match and series ended in a draw.
On Monday, England began the day on 132-3 and like the 2013 match, their primary goal was to play for time, rather than to score runs.
While Dawid Malan fell early, Jonny Bairstow and Moeen Ali were involved in small but fruitful partnerships with Stokes as the visitors ate up time and ticked down the overs.
Stokes and Woakes then defied New Zealand's bowlers for the entire middle session with a patient, low-risk partnership of 83 in almost 31 overs.
The left-handed Stokes, however, chased a short Wagner delivery outside off stump in the final over before the dinner break and only succeeded in hitting the ball to Tim Southee at backward point to be out for 66.
His dismissal left England heading into the final session under lights at 300-7 and needing to survive another 31 overs, but when Astle struck early to dismiss Overton, the victory was in sight.
"We weren't good enough, simple as that really," England captain Joe Root said.
"It's very difficult to get back into the test match after such a poor start. I thought the character of the dressing room and the efforts today were fantastic."
The second match, a traditional red-ball day game, is at Hagley Oval in Christchurch from March 30-April 3.
Ben Stokeschris woakesEngland vs New Zealand 2018new zealand vs england 2018Tim SoutheeTodd AstleTrent Boult
First Published: March 26, 2018, 1:08 PM IST