After the first session was washed out due to rain on the final day, England motored to 296-3 in the middle session after resuming on 183-2.
Bairstow, whose masterful 162 lifted England from a precarious 55-6 in the first innings, played yet another aggressive knock as he registered England's second fastest-ever test half century off just 30 balls.
Bairstow finished off the game in style when he belted off-spinner Michael Bracewell for a four and a six in a superb 44-ball unbeaten knock.
New Zealand had started the day on a high when Tim Southee had Ollie Pope (82) clean bowled in the first over when play resumed after lunch.
But Bairstow once again exhibited his power-hitting prowess as he provided the new captain Ben Stokes and new coach Brendon McCullum a perfect start to their new roles.
England won the first test by five wickets at Lord's and then chased down 299 runs at Nottingham for a five-wicket win in the second test.