The opening goal of the game arrived just nine minutes in when Wilfried Bony fired home from close range after good work from Roland Lamah.
Swansea looked like they were heading for three points and their task was looked to have made somewhat easier late on when Xandao was sent-off for a second yellow card.
However, Kuban threw on Ibrahima Balde - as they did two weeks ago when he won the penalty to deny Swansea at the Liberty Stadium - and he sealed a draw in dramatic fashion.
With just moments left Ivelin Popov played a wonderful ball into the path of Balde and he finished brilliantly over Michel Vorm to deny Swansea, just as they did last month in South Wales.
Swansea's first-half performance deserved so much more, but they paid the price for a poor second-half display, while Laudrup's decision to substitute three of his best players on the night - Bony, Lamah and Dyer - backfired.
Swansea could have led inside 90 seconds after Xandao had cynically cut out a lofted through-ball with his hand, earning him a booking.
Jonathan De Guzman took the free-kick and picked out Bony, but the Ivorian's solid header was well saved by Aleksandr Belenov.
Kuban spurned a fine chance just moments later as Popov's ball fed Djibril Cisse, only for Vorm to keep out his low shot despite slipping as the striker advanced on goal.
Bony made no such mistake with his ninth-minute finish, keeping his calm to put the Welsh club in front.
Lamah should have doubled their advantage in the 20th minute. De Guzman sprung the offside trap with a lovely weighted ball, but the Belgian dragged across the face of goal with his left-footed effort.
Arsen Khubulov and Xandao worked Vorm as the hosts sought a response but the Swans looked very dangerous and Belenov was forced into another fine save as Bony and Pozuelo helped tee up Dyer, with the keeper tipping the winger's curling shot round the post.
However, the visitors were given a reminder of the slender nature of their lead when Khubulov found Popov in plenty of space on the edge of the box - but the Bulgarian rushed his shot and blazed well over.
Swansea made an assured start to the second half, dominating possession and building pressure in the Kuban half - but their good work was almost undone in the 53rd minute.
Ben Davies was robbed of the ball and when it was put into the box it found Cisse on the edge of the six-yard box, but the striker could not control and the chance was gone.
The Frenchman did find the net a few minutes later - but he was rightly flagged for offside.
For the first time Swansea began to look unsettled - not helped by the removal of the trio of Bony, Lamah and Dyer - giving the ball away needlessly and inviting Kuban on to them.
Cisse was continuing to be a thorn in the side of the visitors but he again failed to convert after nipping past Davies and Wayne Routledge.
The Kuban recovery looked to have been halted when they were reduced to 10 men after Xandao was guilty of a woeful tackle on substitute Alvaro Vazquez.
But Swansea were to again be denied by the Russian side as Balde struck with two of the four added minutes gone.
Better news for Swansea came in Switzerland as Valencia's 3-2 win over St Gallen virtually assures Michael Laudrup's men of a place in the last 32.