Alastair Cook's side were bowled out by New Zealand for 167 in the first innings after play was washed out on day one, but managed to salvage a draw thanks to an improved second-innings batting display on a flat pitch.
Knight said England need to address the reasons for their slow start and felt they seemed to take their eye off the ball against a side that is ninth in the ICC Test rankings, seven places below themselves.
He told Sky Sports: "Simon Doull, in interview with Alastair Cook, said it felt like a winning draw but I'd look at it completely the other way and say it was a winning draw for New Zealand.
"Yes, England showed a lot of fight, but the first thing I'd identify is what happened at the start and what went wrong on effectively the first day of this game.
"Alastair Cook said he didn't really know, so there's a bit of soul searching going on there.
"For what it's worth, my take on it - and we've seen it plenty of times with England when they come overseas in the first Test match - is it was a little bit of arrogance and a little bit of complacency.
"They got punished by a side they didn't really feel could trouble them in Test match cricket.
"They won't really want to hear that, but it's the only answer I can come up with for a high-quality side playing against a side that is, let's face it, plenty of ranking points below them.
Knight even saw signs of complacency on day five after England had battled hard to put themselves into a match-saving position.
Centuries from Alastair Cook and Nick Compton, plus a determined stand of 56 from nightwatchman Steven Finn saw the tourists save the game, but the run-out of Joe Root in the final session of the match provided more evidence of England's complacency.
"It was almost as if at tea time that they felt it was almost job done," Knight added.
"We felt this at the start of the game - there was a bit of arrogance and a bit of complacency about the way they started - and it was almost like they thought they'd done the job.
"They'd talked about batting until tea and then there was an injudicious sweep from Steven Finn - and let's not be too critical of Steven Finn because he'd done a wonderful job
"It was almost as if Ian Bell had switched off as well and Joe Root got four and no runs here on the flattest wicket he'll ever play a Test match on.
"It just looked like they'd taken their foot off the gas ever so slightly and maybe that's the lesson moving into the next game.
"However much in ascendancy they are in Test match cricket over this New Zealand side, if they're just off it for 10 minutes, half an hour or an hour then they're going to get punished."
New Zealand's seamers toiled for the final two days without much success and Knight said the nature of the wicket in Dunedin helped England's cause significantly.
He added: "When you're bowled out for 167 in the first innings of a Test match then generally you don't save that. However good or bad the opposition are, a Test match side will make you pay for that.
"But our point out here, having seen the pitch for the last four days, is there's been very little for the seamers and absolutely nothing for the spinner.
"If you judge a Test match pitch, then by the time you get to day three-and-a-half or day four then you're expecting to see a little bit of deterioration.
"We could still have had an excellent game if we'd had five days of cricket, but in general there are too many pitches that we see that don't do enough come days three, four and five."