The Shaymen matched their Unibond Premier Division hosts stride for stride in the opening 45 minutes and turned round level at 1-1.
But they subsided after the break as Nantwich took full advantage with three goals in 28 minutes.
"You expect your defenders to deal with situations and they didn't," said Vince. "That's why we are not in the next round of the Cup.
"Nantwich are a league above us and should be better than us, but if we had defended properly on Saturday we would have been in the next round.
"And that is not the back five it is the whole team. We did not close down quickly enough, we didn't deny them space enough."
Town took a 10th minute lead at the Weaver Stadium through Peter Moore, but Nantwich levelled six minutes later and bagged three second half goals as they eased to victory.
"No one expected us to win apart from ourselves," said Vince. "I genuinely thought we were going to do well and that we would beat them.
"You can only beat teams if you defend well and we conceded four goals, with bad defending for every one of them."
Vince was particularly disappointed because all four of Nantwich's goals were preventable.
"The first goal came after we failed to track the player who came from the inside," he said.
"We were in the game at half time and there was not much between us.
"But in the second half we conceded a goal from a basic error at a corner, their player scoring from six yards because no one tracked him.
"You have to defend set plays"
"They had more possession than us, but we were still in it then Lincoln gave away a stupid penalty and it killed us.
"For the last one Lincoln and Tony Barras didn't attack the ball."
Town suffered a major blow before half time in Cheshire when they lost in-form Paul Gedman to injury.
And they never really threatened after the man who had scored five goals in his first three Town starts had left the field.
"It was a big blow to us," said Vince. "He was training on Thursday with Droylsden and picked up an injury.
"It is disappointing because he should have been with us, but he could have picked up an injury if he had been with is.
"Ashley wasn't really a threat after Gedman went off. He didn't do anything for me.
"You have to look at the service, but when the ball is there you have to be better with it.
"Their forwards were better than ours because they got behind the defence.
"We played everything in front of their defenders."
"We were better than them in the first half, but we are not performing in the second half of games.
"Our ball retention in the second half was not good and our defending was poor.
"We had one chance in the second half from a Steve Payne header."