Both sides had their chances in the first half and there were screams for a penalty from Celtic when Reo Hatate went down in the County box under pressure from Eli Campbell, but play was waved on and the video assistant referee (VAR) was happy to let the action continue.
Kyogo fired Celtic into the lead minutes after missing a gilt-edged chance as he poked in a lovely cross from right-back Alistair Johnston.
The Japan forward was flying as the half drew to a close, forcing County goalkeeper Jordan Amissah into an incredible save.
Johnston's foul on County debutant Kieran Phillips gifted the hosts a second-half penalty, which was dispatched by White for the equaliser.
Furuhashi fired a bullet header at Amissah who, despite getting a hand to it, could not keep the ball out of the net as Celtic regained the lead in the closing stages.
Engels' penalty - awarded following a VAR review of James Brown's challenge on Hyun-Jun Yang - was saved by Amissah before the spot-kick was retaken because the County keeper had strayed well off his line.
Engels stepped up again and fired into the opposite corner, then McCowan made it 4-1 when he popped the ball into the bottom corner.