The All Blacks held their nerve in a brutal defensive contest in Dunedin in which both teams scored two tries, extending their winning run on home soil against England to eight matches since 2003.
It took a grinding effort in the second half to secure victory after England winger Immanuel Feyi-Waboso scored a try to break a 10-10 deadlock at the interval.
The All Blacks struggled to penetrate a disciplined England defence and never threatened a try in the second half, but they did enough to set up McKenzie for penalty goals in the 54th and 65th minutes.
McKenzie was timed out when lining up a third penalty attempt in the dying stages but England failed to score from the final play.
The final whistle came as a relief to new All Blacks captain Scott Barrett, who admitted the game hung in the balance over the closing stages.
"Immensely proud, to be honest it could have gone either way but I guess we rode our luck," Barrett said.
"They shifted us around, those little kicks in behind turned us around and squeezed us at times.
"We were a bit slow to adapt and we got a couple of penalties just to nudge us in front."
The All Blacks came into the game after just 10 days under former Canterbury Crusaders mentor Robertson, who succeeded Ian Foster after the World Cup.
They had also lost a handful of experienced players - retired or playing abroad - in the wake of their World Cup final loss to South Africa.
"Got into a hell of an arm wrestle but we know Steve Borthwick's teams are very good at that," said Robertson. "It was a grind but we got there."
Coming off a 52-17 win over Japan two weeks earlier, England played with great cohesion but also struggled to create try-scoring chances.
Skipper Jamie George said his side will take confidence into the final Test in Auckland next Saturday.
"A one-point game makes it exciting for next week," George said.
"We loved defending, we wanted to put the All Blacks under pressure. We want to back that up next week. It was a close game of fine margins."
Neither team led by more than five points throughout an absorbing contest dominated by the physicality of the collisions and some confusion on rulings made by Georgian referee Nika Amashukeli.
England fly-half Marcus Smith missed an early penalty before All Blacks winger Sevu Reece opened the scoring with a try after fielding a McKenzie cross-kick.
The visitors struck back through a close range try by lock Maro Itoje but New Zealand regained the ascendancy when 2023 World Player of Year Ardie Savea capitalised on a break from full-back Stephen Perofeta to score in the corner.
New Zealand conceded a penalty after the half-time hooter which was slotted by Smith to square the scores.
England dominated the opening exchanges after the break and created space for Feyi-Waboso to cross in the corner.
England prop Joe Marler could be in doubt for the second Test at Eden Park after exiting early with a foot complaint while All Blacks scrum-half TJ Perenara didn't return from the half-time break following a knee injury.
