Lorenzo Melgarejo was on target twice, opening the scoring in the third minute and getting the third 18 minutes from time.
Vladislav Ignatiev netted the second in the 55th minute and Charles Kabore rounded off the rout at the death as Kuban moved above their opponents into third place in the pool on five points.
The comprehensive win was no less than they deserved for a display of total domination and indeed the scoreline flattered St Gallen.
The hosts got off to a flying start as Melgarejo volleyed Anton Sosnin's ball in from the right low into the net.
Aleksei Kozlov forced a parry from St Gallen goalkeeper Daniel Lopar with a shot from the centre of the area, while at the other end Aleksandr Belenov had to be alert to deny Mario Mutsch.
That was a rare moment of promise from the Swiss side, though.
The Russians were well on top and Melgarejo almost got a second only to head wide from close range from Sosnin's free-kick.
Djibril Cisse fired against a post from outside the area and Ignatiev also hit the frame of the goal with an effort from much closer in.
Kuban were desperately unlucky not to be more than one goal ahead at half-time, but they did not have to wait long into the second half to double their lead.
A minute after Ivelin Popov had a shot parried by Lopar he laid on the pass which Ignatiev slid home from the left of the six-yard box.
Melgarejo made it three with his second of the match in the 72nd minute with another close-range finish, this time pounding on Igor Armas' knock-down.
Already home and dry, the hosts nevertheless kept on pressing and Cisse headed over when well-placed.
But Kabore was on target from another Armas head down, firing a low shot into the right corner in injury time to round off a dominant display.