New Zealand achieved a resounding victory at Auckland when they won the first of the two Test matches. England faced a humiliation when they were bowled out for 58 in the first innings. While Trent Boult bagged six wickets then, Southee bagged four.
The visitors were dismissed for 320 early in the final session under lights on the final day of the day-night match at Eden Park, giving New Zealand only their 10th Test victory against England.
"We weren't good enough, simple as that really," Root told reporters. "It's really difficult to get back into a Test match after such a poor start. We have to learn some lessons quickly... and we know that we have to adapt better."
England have failed to win a Test away from home since they beat Bangladesh by 22 runs in Chittagong in October 2016 and have lost 10 of their last 12 tests overseas, including 4-0 series defeats to India and Australia.
Root, however, said the second innings performance had showed the team were still fighting hard to turn that series of defeats around.
"I thought the character of the dressing room and the efforts today were fantastic," Root added.
"You go into a day like today and ask the guys to show how much it means to them to play for England and it was an example of the guys' determination to represent the country.
"I think we can take a lot from the second innings and make sure that we are better."
The second test, a traditional red-ball match, is at Hagley Oval in Christchurch from March 30-April 3.
Earlier prior to the Test series, England won the 5 match ODI series against New Zealand 3-2.