Michael Laudrup's side had been made to work hard all night at the Liberty Stadium but it looked like they would emerge as winners thanks to Michu's goal in the 68th minute.
However, goalkeeper Gerhard Tremmel brought down Ibrahima Balde and former Liverpool striker Djibril Cisse stroked home the spot-kick with virtually the last kick of the game.
The result leaves Swansea just one point ahead of Valencia at the top of the table, although they have a four-point cushion over third-placed St Gallen.
Swansea knew a win here would give them one foot in the knockout stages, but there was little sign of any urgency during a stuttering display.
Such was Swansea's lack of fluency, it took them 19 minutes to test Aleksandr Belenov, and then it was a tame effort from Nathan Dyer after a ricochet had helped the winger past two defenders.
A clearer chance followed seconds later as Wilfried Bony and Michu combined to pick out Jonjo Shelvey unmarked, but the midfielder snatched wildly at his shot which flew hopelessly wide.
Swansea were also experiencing a few scares of their own at the other end.
Chico Flores managed to steer an Arsen Khubulov pass over his own bar, just about, after a simple lofted ball from Charlos Kabore had found the hosts at sixes and sevens.
Dwight Tiendalli then had a lucky escape as, having realised his header back to Tremmel was short of power, pulled down Lorenzo Melgarejo, with the fact the incident occurred in a wide channel possibly saving the Dutchman from dismissal.
Another hopeful Kabore punt forward soon found the code to breach Swansea's defensive security again on the half-hour, but Popov dallied and then produced a clear dive under pressure from Neil Taylor as the chance was wasted.
Swansea simply had to improve after the break, but, aside from a rasping Shelvey effort which struck Igor Armas straight in the face when appearing to be bound for the top corner, the visitors were not unduly troubled.
However, as the half wore on Swansea gradually began to take a grip on proceedings, and Michu got the opener.
Alejandro Pozuelo scooped an exquisite pass forward for Michu following a slick move and he beat the keeper to prod home.Swansea looked calmer after taking the lead but could not kill the game and Shelvey's wayward pass allowed Kuban to launch their critical last-gasp attack.
|Swansea City||Team Statistics||Kuban Krasnodar|
|0||1st Half Goals||0|
|5||Shots on Target||3|
|7||Shots off Target||11|
|72.1||1st Half Poss.||27.9|
|61.9||2nd Half Poss.||38.1|