Defeat means Celtic remain six points behind Scottish Premiership leaders Hearts – albeit with a game in hand – but Nancy has urged the fans to “keep faith”.
His side seemed to have responded positively to Sunday’s League Cup final defeat by St Mirren with a dominant first-half display, gilded with Daizen Maeda’s goal.
However, Celtic missed a succession of chances and could not quell a revived United after the break, with stunning strikes from Krisztian Keresztes and Zac Sapsford turning the game in the struggling hosts’ favour.
It ended eighth-placed United’s seven-game winless run and their 11-year, 24-match wait for a win over Celtic.
“We lost the game in three minutes – two set-pieces that we conceded,” Nancy told BBC Scotland. “Maybe the second one was offside before the corner.
“I think we did a good job – I think we had a good performance with all the opportunities we had in the first half. If we score the two or three tap ins we had, it would have been a different score.”
Nancy said he recognises that the ongoing turmoil is “difficult” for the Celtic fans, but the 48-year-old is adamant that he can see improvements.
“I can understand,” he said. “I am a human being and disappointed and the connection with the fans is important for me. But we had clear chances and we have to work on certain things.”
Asked if he regretted changing a system that proved succesful for interim manager Martin O’Neill – who won seven of his eight games in charge – Nancy added: “Martin plays a certain way and I want to see something that can win games for the future.
“At the moment, it is not that, but I can see we are improving. We’ve lost four games in a row, but today we were close to winning the game.
“It’s the direction we have to keep going in, and we have to be stronger to avoid conceding these kinds of goals.”