United looked set to record their 12th League One win since August when Duffy, applying the finishing touches to a well-worked set-piece routine, handed the visitors a first half-lead.
Although the result stretched their unbeaten run in the competition to 15 games, Wilder accused United of “throwing away” two points following Patrick Bauer’s last gasp strike for Charlton Athletic.
“The manager has been in the game a long time and he knows when to put the hammer down,” Duffy said.
“And we deserved it, he had every right to have a ‘go’ at us afterwards.
“We’ll be sitting on that coach on the way home gutted. We know where we want to be and I think we’re good enough to get there.
“Yes, we can be harsh on ourselves but we set our standards really high and, if we keep hitting that bar, then we won’t go far wrong.”
United appeared on course to do exactly that when, after keeping the dangerous Josh Magennis and Ademola Lookman at arms length during the early skirmishes, seized control midway through the opening period.
Leon Clarke struggled to impress on his return to the starting eleven but was responsible for winning the free-kick which, after making a decoy run, Duffy converted following Paul Coutts’ slide rule pass.
With Coutts, Duffy and John Fleck stamping their authority on proceedings, that should have been the cue for United, now third in the table, to establish an unassailable lead.
But with Dillion Phillips denying Fleck, substitute Caolan Lavery and Billy Sharp after the break, Charlton levelled when Bauer turned home from close-range after meeting substitute Adam Chicksen’s centre.
“We want to go to any ground in this league and win,” Duffy said. “We’re always confident we can dominate the game. We’ve done that from start to finish, pretty much for 90 per cent of the game, so the gaffer has had a go at us.
“He’s told us we need to be more ruthless and have that desire to put the ball in the back of the net. Not just the strikers, all of us, even the lads at the back.”
Alan Knill, United’s assistant manager, has a gift for devising ingenious set-pieces.
Charlton boasted the fourth best defensive record in the division ahead of this game but were utterly bamboozled by the chain of events which ended with Duffy threading an angled drive past Phillips.
The former Birmingham City player joined Coutts, Daniel Lafferty and Fleck in a huddle following Bauer’s foul on Clarke before peeling away and receiving the ball.
As Charlton’s rearguard tried to fathom what was going on, he picked his spot before wheeling away in celebration and gesturing towards Knill on the bench. Although, as Duffy later revealed, this particular piece of genius had a twist.
“I’m supposed to square that across the box,” he said.
“But I saw the goalkeeper edge out, go a little bit too early, so I just thought I’d have a go. They thought we were going for the one we did at Scunthorpe. They were shouting ‘look out for the man on the edge.’
“But they didn’t know that one did they? Fair play to Al, he’s got tons of those up his sleeve. So there’s plenty more of them that people haven’t seen yet.”
Charlton’s owner Roland Duchatelet has published a book entitled “The Road to More Net Domestic Happiness” but there was precious little cheer, until Bauer’s dramatic intervention, inside this famous old stadium on Saturday afternoon.
With Karl Robinson, the Belgian’s seventh managerial appointment since his takeover in 2014, watching from the stands before being officially unveiled today, the home supporters made their feelings known by throwing miniature footballs and taxis onto the pitch. Referee Graham Scott was forced to call a halt to the game in the third minute until it was cleared.
It is a measure of the progress United have made under Wilder this season that drawing with Charlton could provoke such a furious response from the 49-year-old.
But his reaction was understandable. The hosts entered this game having scored seven goals in their last two outings and Magennis, searching for his third in three, burst out of the blocks before Fleck and Coutts cut the Northern Irishman’s supply line.
From that moment on, United grew in stature and Charlton became increasingly fractious.
Sharp saw an overhead kick saved after Duffy’s opener and Fleck also drew a fine block from Phillips.
The goalkeeper excelled himself again when Duffy, after two clever feints, unleashed a vicious shot .
But United’s failure to translate possession into points, was not solely down to his heroics. Chris Basham should have done better after dancing through Charlton’s rearguard before dragging wide while, on too many occasions, their final pass went awry.
“Their celebrations at the end were like they’d just won,” Duffy said.
“Which I think tells you a lot. We just feel really disappointed not to have done more.”