Kuban, who rescued a point in their Group A clash in Wales last month thanks to Djibril Cisse's injury-time penalty, repeated the dose in Russia thanks to Balde's composed finish.
Swansea seemed set for victory on Thursday thanks to Wilfried Bony's early strike, with Cisse missing a host of chances to level for the hosts and also seeing an effort ruled out for offside.
Brazilian defender Xandao was sent off with 15 minutes remaining, further harming Kuban's hopes, but Balde reacted quickly to a flick from the influential Ivelin Popov in the second minute of stoppage time.
The Senegal international, who came off the bench in the 74th minute, calmly lobbed the ball over onrushing Swansea goalkeeper Michel Vorm as Michael Laudrup's side were again denied in the dying stages.
Valencia's subsequent victory over St Gallen means the Spanish side replace Swansea as group leaders, while Kuban are still bottom with two points.
Bony, Roland Lamah and Ben Davies headlined Swansea's seven inclusions, while Kuban also fielded a much-changed team, with Cisse one of five alterations.
Swansea went close as early as the second minute, Bony's header forcing Kuban goalkeeper Aleksandr Belenov into a smart save, while Cisse could only shoot straight at Vorm when played through by Popov at the other end.
The match's frantic start continued as Bony fired Swansea in front in the ninth minute, converting Lamah's cut-back from close range after a clever team move that also involved Alejandro Pozuelo.
Vorm was tested by both Charles Kabore and Xandao in the aftermath and the end-to-end nature of the contest did not relent as Lamah spurned a chance to give Swansea a two-goal lead, shooting wide with his left foot.
Kuban were often restricted to attempts from distance, with Roman Bugaev and Popov missing the target, while Vorm dealt with Arsen Khubulov's volley comfortably.
Dyer forced a stunning save from Belenov with a curling shot from 20 yards as the first half drew to a close, while after the resumption Cisse mis-kicked when presented with a great chance by Popov.
Cisse did score in the 57th minute, only to have it disallowed, and Kuban began to control the match as Popov shot at Vorm and Cisse fired wide.
They lost momentum when Xandao was sent off, the defender, who was initially booked for a handball inside the first 45 seconds, given his marching orders after going through the back of Swansea substitute Alvaro Vazquez.
However, there was still one last chance for the hosts and Balde's coolness under pressure ensured that both sides finished with a point.