The momentum seemed to be with the hosts but it all changed in the 110th minute when a hopeful through-ball bounced off Jonny Evans and fell kindly for Duris to beat Bailey Peacock-Farrell at his near post.
The Burnley goalkeeper will feel he could have done better, but it was a bitter blow for Northern Ireland as their hopes of qualifying for a second consecutive European Championship were quashed.
Ian Baraclough's men began the night in the unusual position of favourites against a side who sacked their manager last month, but they could not capitalise as Slovakia made the most of the chances they were gifted.
Northern Ireland, so often guilty of making a slow start in recent times, looked determined to put that right as they raced out of the blocks to peg Slovakia back in the early going, roared on by the 1,060 fans allowed into Windsor Park.
But finding a final ball proved elusive, and Slovakia had already threatened through ex-Manchester City man Albert Rusnak before they took the lead against the run of play.
George Saville tried to chest the ball back to Evans from the halfway line but noticed too late that the Leicester defender had stepped up, allowing Kucka the freedom to race through and fire beyond Peacock-Farrell.
Northern Ireland responded well to the setback, however.
Niall McGinn, restored to the starting eleven for the first time since scoring in the play-off semi-final against Bosnia and Herzegovina last month, almost caught out Fulham goalkeeper Marek Rodak with a cross that drifted towards the far post, before Stuart Dallas dragged a shot wide.
McGinn's cross narrowly eluded Josh Magennis before the Aberdeen winger fired over from a loose pass by Stanislav Lobotka, and Northern Ireland trailed at the break.
They were straight back on the front foot in the second half, using a high press to keep the visitors under pressure around their own box, with Conor Washington testing Rodak after McGinn won the ball.
But the occasion was perhaps getting to Northern Ireland as they began to rush passes, allowing Slovakia to assert increasing control.
Gavin Whyte and Jordan Thompson replaced Washington and Saville, but it was the decision to send on Kyle Lafferty and Liam Boyce in place of McGinn and Magennis that altered the dynamic.
The leveller came with two minutes left. Paddy McNair raced into the box from the right, looking to feed substitute Boyce, but it was Skriniar who turned the ball into his own net.
Lafferty then fired against the post as Northern Ireland sought an unlikely winner before the 90 minutes were up, but instead the game went into extra time.
Their efforts began to take a toll. Craig Cathcart went off holding his groin in the 97th minute, while McNair followed soon after, suffering from cramp.
The energy faded from Northern Ireland as they went, and the hammer blow came 10 minutes from time as Duris struck.
Inevitably there was a late chance to level, with Thompson's cross picking out Evans six yards from goal, but the exhausted defender could not find the power needed to beat Rodak.