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 games 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 reprieve 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.
Source: PA
Source: PA