Michu opened the scoring with his fourth goal in Europe this season, but Swans keeper Gerhard Tremmel gave away a spot-kick which was ruthlessly dispatched by the French striker
An otherwise dull contest sparked into life late on, as the Frenchman converted from the spot after Michu had combined brilliantly with compatriot Alejandro Pozuelo to open the scoring midway through the second half.
Swansea made six changes from the team that thrashed Sunderland 4-0 on Saturday, with Gerhard Tremmel and midfielder Jonjo Shelvey coming in, while Kuban manager Viktor Goncharenko recalled striker Cisse in one of his four alterations.
Goncharenko – taking charge of just his second match since replacing sacked Romanian Dorinel Munteanu – would have been pleased with his side's start but it struggled to create clear-cut opportunities.
And after Nathan Dyer forced Kuban goalkeeper Aleksandr Belenov into a smart save with a testing left-footed effort, Shelvey nearly gave Swansea the lead in the 21st minute.
The former Liverpool man combined well with Wilfried Bony and Michu before producing a good first touch, but shot wide from inside the penalty area, before Kuban enjoyed a terrific five-minute spell.
First Arsen Khubulov was released down the right, but saw his teasing cross – that had Cisse waiting to pounce – somehow turned over the bar by a desperate Chico Flores, before Maksim Zhavnerchik's 25-yard effort was pushed wide by Tremmel.
Shelvey fashioned another good chance for himself, after lifting the ball over Kuban defender Igor Armas, but he again shot wide, before Dwight Tiendalli was lucky to stay on the pitch.
The Swansea defender brought down Lorenzo Melgarejo when he looked to be through on goal, but referee Kristinn Jakobsson only produced a yellow card.
Cisse and Ivelin Popov had half chances for Kuban shortly after the break before Swansea boss Michael Laudrup introduced Spanish pair Alvaro Vazquez and Angel Rangel in an attempt to change the match.
It was another Spaniard who gave Swansea the lead, however, as a patient passing move saw Michu net his fourth goal of the Europa League campaign to break the deadlock.
The striker was involved in the build-up, receiving the ball from Shelvey before playing in Pozuelo, and his clipped pass over the top exposed the Kuban defense, with Michu first to the ball as he stabbed past the on-rushing Belenov.
Swansea sat back after taking the lead, but was stung on the break with the last kick of the game. Kuban substitute Ibrahima Balde raced through on goal and nicked the ball past Tremmel, who brought down the Senegalese forward, and Cisse stepped up to dispatch the resulting penalty with aplomb.