Stokes made a patient 66 from 188 balls as he and Chris Woakes (52) helped the tourists recover after early wickets.
The Durham man's dismissal three balls before the dinner break effectively ended England's hopes of a draw.
They will get a chance to level the two-match series at Christchurch in the second Test, which starts on Thursday.
The seeds of defeat were sewn on day one of this match when England were dismissed for a paltry 58 - their sixth-lowest total in Test history and arguably the batting nadir of the winter.
However, with two days severely impacted by rain, they had a realistic chance to salvage a draw if they could bat out the final day.
This was not a typical day five pitch at Eden Park, and conditions were relatively comfortable for batting but the task proved beyond England.
They lost three wickets in the first session, but Stokes proved difficult to dislodge as he shared a seventh-wicket stand of 83 with Woakes.
His vigil was ended three balls before the end of the second session, though, when he skied a bouncer from Neil Wagner (3-77) and was caught by Tim Southee in the point region.
That left New Zealand requiring three wickets in 31.3 overs under the floodlights and Wagner duly had Woakes caught at short leg before James Anderson was last man out when he was caught by Trent Boult off Todd Astle, who finished with 3-39.