New Zealand had fought back through its two prolific run-getters in the series -- Tom Blundell (88 not out) and Daryl Mitchell (56) -- but the top order's struggle in the series let down the Kiwis.
Nicholls, on the stroke of tea, drove hard at Leach only for the ball to ricochet off non-striker Daryl Mitchell's bat and loop gently to Alex Lees at mid-off.
Ben Stokes said that England's victory push had started when his bowlers came back to bowl in the second innings and managed to restrict the Blackcaps below 300.
Set 299 to win in a little over two sessions on Day 5, England reached its target in 50 overs with Bairstow falling for 136 off 92 balls just before the victory was sealed.
England batter Joe Root completed 10191 Test cricket runs and surpassed former India batter Sunil Gavaskar's record of 10122 runs.
Nasser Hussain said that a score in excess of 500 could dent any team's confidence but seeing Ben Stokes' side match New Zealand with the bat was heartening.
England vs New Zealand second Test: Hope Joe Root's charisma will carry us to victory, says Ben Foakes
The second Test at Trent Bridge looks set for a thrilling end on the last day after England bowlers restricted New Zealand to 224/7.
Root, currently the world's most in-form batter in test cricket, was still there at stumps on 163 not out to give England hope of coming out of the first innings with an unlikely lead.
Nasser Hussain opined that England skipper Stokes will need at least one of his top three to go big at Trent Bridge.
England, set 277 to win, finished on 279 for five, with former captain Root 115 not out after the hosts had slumped to 69 for four.
New Zealand on Wednesday named an extended 20-man squad for the tour, designed to cover the early absence of players involved in the Indian Premier League.
Boult said he has been struggling to forget the 49th over in which he took a catch of Ben Stokes off Jimmy Neesham but ended up stepping on the boundary to let the advantage slip.
Rashid stated that Morgan knows his game inside out and has been with him through 'good and not so good times'.