Eddie Jones's team roared into a 15-0 lead after scoring two tries at a rain-swept Twickenham but the world champions held their nerve to eke out the narrowest of victories.
England could have won if Sam Underhill's late try had not been ruled out by the TMO for offside, the verdict going against the hosts a week after they had got the benefit of the doubt from the officials to hang on for a 12-11 win over South Africa.
Overall, though, it was an excellent display by England, missing several first-choice players due to injury, and Dallaglio was impressed.
"England's players will remember this as the day they got themselves properly back on track," he wrote in the Sunday Times.
"England were magnificent, barely recognisable from the team who beat the Springboks a week ago," he added. "Fifteen points clear after 25 minutes and it was hard to remember when England has played this well."
It was quite a turnaround for Jones's team, who had lost five successive tests in a dismal run this year and were being written off as genuine contenders to win next year's World Cup.
But back to back victories over South Africa were followed by a courageous performance against the All Blacks in which several players excelled, including lock Maro Itoje, flyhalf Owen Farrell and 22-year-old flanker Underhill.
"Maro Itoje was excellent, a talismanic figure in the pack and constantly in the face of the All Blacks," Dallaglio said.
"Farrell was his aggressive and influential self. He doesn't have off days."
Underhill worked hard and delivered several trademark chop tackles before his mazy 30-metre sprint down the left wing bamboozled New Zealand flyhalf Beauden Barrett and would have gone down in Twickenham folklore had it not been chalked off.
"I also liked the way Sam Underhill played," Dallaglio said. "He brings so much energy to the team and his eagerness to make tackles and take the ball on was infectious."
England's lineout did malfunction after the interval as New Zealand took control of possession.
"Different factors explain why it went so badly in the second half because the conditions made it hard for replacement hooker Jamie George and his jumpers but more important than that was the difference in New Zealand's lineout defence," Dallaglio said.
Jones was rightly proud of his team's effort and they will go into the next two tests against Japan and Australia on Nov. 17 and 24 brimming with confidence.
(Editing by Clare Fallon)
