Swansea were again pegged back by a late Kuban Krasnodar goal as Ibrahima Balde's stoppage time equaliser held them to a 1-1 draw in Russia.
Wilfried Bony had put the Swans ahead in the ninth minute and they looked set to move to the brink of a place in the knockout stages of the Europa League when their hosts had defender Xandao sent off with 15 minutes remaining.
But, as had happened when Djibril Cisse struck a stoppage time penalty in Swansea two weeks ago, they could not claim all three points as Balde just beat Michel Vorm to a lofted pass and lifted the ball over the goalkeeper to make the final score 1-1.
Swansea have now won just two of their last nine in all competitions, but they should have left the Kuban Stadium with three points after dominating the first half.
Bony could have given Swansea the lead in the opening seconds, directing a powerful header towards the near post, only for Aleksandr Belenov in the Kuban goal to make a sharp save.
But there was no repreive for the hosts when excellent link-up play between Alejandro Pozuelo and Roland Lamah, who impressed in only his third Swansea start, teed up Bony.
Lamah surged into the area and, when Pozuelo could not control, Bony took over to fire convincingly beyond Belenov.
Lamah and Nathan Dyer had chances to extend the lead, with the former dragging wide when played in by Jonathan de Guzman, and Belenov saving from the latter.
The failure to extend the lead almost proved costly as Djibril Cisse had a goal ruled out for offside, and failed to control at close quarters, but Kuban's charge was dented when Xandao saw red for a second yellow card.
However, Ivelin Popov's ball into the area saw Balde nip in ahead of Vorm to earn the hosts a share of the spoils and keep Swansea waiting to secure their spot in the last 32.
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.
De Guzman took the free-kick and picked out Bony, but the Ivorian's solid header was well saved by Belenov.
Kuban spurned a fine chance just moments later as Popov's ball fed 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.