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The Swans were denied a third straight win in Group A as Kuban Krasnador snatched a point from the penalty spot with virtually the last kick of the game.
Michu's second-half strike looked to have been enough to secure the Swans victory but former Liverpool striker Djibril Cisse claimed a last-gasp draw after sub Ibrahima Baldé was brought down by Gerhard Tremmel.
After a goal-less first half, Michu tucked home his sixth goal of the season - a fourth in Europe - as he beat keeper Aleksandr Belenov to Alejandro Pozuelo's flicked pass in the 68th minute.
But late drama at the Liberty saw Cisse keep his cool from the spot to claim a point for the visitors.
Laudrup's side remain top of Group A on seven points, with Valencia on six points and St Gallen on three while Kuban moved onto one.
The Swans made six changes to their starting XI.
Michael Laudrup opted for Gerhard Tremmel in goal while left-back Neil Taylor came in to captain the side for the first time with Dwight Tiendalli starting on the opposite flank.
In midfield, Jose Canas and Jonjo Shelvey got the nod in place of Leon Britton and Jonathan de Guzman while Alejandro Pozuelo took the place of Wayne Routledge out wide.
Missing through injury for the Swans were trio Ashley Williams, Ben Davies and Pablo Hernandez.
SWANS: Gerhard Tremmel, Dwight Tiendalli (Angel Rangel, 59), Chico Flores, Jordi Amat, Neil Taylor (capt), Jose Canas, Jonjo Shelvey, Michu, Nathan Dyer (Roland Lamah, 77), Alejandro Pozuelo, Wilfried Bony (Alvaro Vazquez, 59).
SUBS: Michel Vorm, Garry Monk, Leon Britton, Jonathan de Guzman.
KUBAN: Aleksandr Belenov (capt), Igor Armaş, Xandão, Djibril Cissé, Charles Kaboré, Maksim Zhavnerchik, Artem Fidler, Aleksei Kozlov, Lorenzo Melgarejo (Gheorghe Bucur, 63), Ivelin Popov (Ibrahima Baldé, 76), Arsen Khubulov (Vladislav Ignatyev, 68).
SUBS: Eduard Baychora, Ángel Dealbert, David Tsoraev, Artur Tlisov.
In what proved an even start, the visitors threatened early on with centre-back Igor Armaş backheeling his volley over the crossbar from a corner before Djibril Cisse then volleyed wide from a cross.
The Swans hit back via Nathan Dyer, who fired straight at keeper Aleksandr Belenov, and Jonjo Shelvey as the midfielder misdirected his shot after neat link-up play by Wilfried Bony and Michu.
Kuban then created the best opportunity of the game so far as Arsen Khubulov latched onto a through ball and he sped down the visitors' right flank before seeing his cross intended for Cisse superbly cleared over his own crossbar by Chico Flores.
Gerhard Tremmel then had to be alert as he guided a dipping Maksim Zhavnerchik effort wide for a corner as the visitors showed their intent.
At the other end, Shelvey saw his left-footed half-volley curl wide after he latched onto Dwight Tiendalli's chipped pass before Bony volleyed high and wide after neatly controlling captain Neil Taylor's cross.
The end-to-end action continued, with Tiendalli earning a booking after he took down Ivelin Popov as he attempted to take advantage of the Dutchman's weak passback.
Shelvey then rifled a 25-yard free-kick over the crossbar after Michu was hauled down by Armaş, which earned the defender a booking.
HALF-TIME: Swans 0 Kuban Krasnodar 0
Both sides started the second half slowly, and Laudrup looked to inject a fresh approach to his side's game on the hour mark as Angel Rangel and Alvaro Vazquez came on in place of Tiendalli and Bony.
Minutes later, Kuban made a change of their own, Gheorghe Bucur on in place of Lorenzo Melgarejo.
Then, on 68 minutes, came the moment the home crowd had been waiting for.
Shelvey found Pozuelo, and the Spaniard flicked a pass over the Kuban defence for Michu, who won the race with keeper Belenov to poke home.
The Swans immediately went in search of another goal, Shelvey seeing his free-kick padded down by Belenov before Vazquez's attempted chip was snuffed out by the keeper.
Laudrup made a final change in the last 15 minutes as Roland Lamah came on in place of Dyer.
Then came the heart-breaking moment as Tremmel and Ibrahima Baldé were locked in a race for the ball in the box, and the substitute went down under the challenge.
The penalty was awarded, and Cisse coolly slotted into the bottom corner with virtually the last kick of the game.