Hand, who spent four years in Sheffield with the Steelers prior to joining Ayr Scottish Eagles in the summer, netted a brilliant equaliser five minutes from time. It proved a remarkable comeback for Hand, who has been five years in the international wilderness after opting out of playing for Britain for a variety of personal reasons.
The Ukrainians enjoyed the early advantage and scored in the 14th minute through Valentyn Oletskiy, but Britain produced a storming second session, deservedly pulling level through Mike Harding of Manchester Storm after 26 minutes on his international debut.
Ukraine regained the lead eight minutes into the final period courtesy of some sloppy netminding from Stevie Lyle, who let down a superb penalty killing performance on the night from the British team.
The Ayr Scottish Eagles defenceman Scott Young had been called on a two-minute cross-checking penalty, and there seemed little danger of Britain conceding with just 11 seconds to kill on the powerplay, but Oleg Polkovnikov fired in a tame shot from just inside the blue line which Lyle inexplicably let past him between his skate and the pipe.
Britain then brilliantly killed a five-minute powerplay after Darren Hurley had received a game misconduct penalty for flooring Kostantyn Butsenko, who lay prone on the ice for several minutes before being carried off. It proved the turning point as Britain came out fired up and Hand netted the game-saving marker.
In the other game, Latvia beat Kazakhstan 6-3, with the game all over inside the opening 30 minutes, by which time the Latvians were 4-0 clear.