Captain Cook batted all day to reach 153 not out, his 27th test century, and Stokes provided the fireworks by smashing a hundred off 85 balls, the fastest ever made in a test at Lord's.
Joe Root contributed 84 and England, leading by 295 runs, will start the final day of a fluctuating match as favourites to take a 1-0 lead in the two-match series.
"It is a great day personally but more importantly we are in a good position as a team," Stokes told the BBC.
"I got close in the first innings and was nervous but to get over the line was brilliant."
England, resuming on 74 for two, lost Ian Bell to the third ball of the day, caught by wicketkeeper Tom Latham aiming a loose drive at Tim Southee.
But Cook and Root dug in and the captain, 32 not out overnight, brought up his fifty off 122 balls.
The New Zealand pace bowlers beat the bat a few times but Root remained typically unruffled and England took lunch on 175 for three.
Root peppered the boundary after the interval and was closing in on the century he missed by two runs in the first innings when he hooked Matt Henry to Trent Boult on the long leg boundary, ending a fourth-wicket partnership of 158.
Cook drove Henry straight down the ground for three to reach his century, his second in two tests after nearly two years without one in international cricket, raising his bat to the dressing room as the crowd gave him a standing ovation.
Stokes, who made 92 in the first innings, struck a few powerful blows before rain forced the players off for an early tea with England on 261 for four.
The floodlights were switched on at a gloomy Lord's and Stokes launched a withering assault on the New Zealand attack.
He reached his fifty off 57 balls before pulling Southee for three sixes over the leg-side boundary.
The left-hander also flayed 15 fours all round the ground and a single off Henry took him to three figures, Stokes punching the air, raising his bat to all corners of the ground and embracing Cook as Lord's applauded his stunning knock.
He only added one more run before he edged spinner Mark Craig to Ross Taylor at slip and Jos Buttler, on 14, nicked Henry to Latham.
New Zealand's misery was not complete, however, Cook going past 150 in the company of Moeen Ali (19 not out) and the pair will resume in the morning looking to extend England's lead.
(Editing by)
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