The Danes believed they had earned a penalty early in the second half when Nico Elvedi was shown a yellow card for bringing down Kasper Dolberg. However, after a VAR review, the referee altered his call to a free kick just outside the box and a red card for the Swiss defender.
Jonas Wind had the ball in the net in the 72nd minute but the goal was chalked off for offside, and nine minutes later the 19-year-old Dorgu was introduced for his first cap as his side chased a winner.
A minute later he was on the scoresheet, latching on to a ball from fellow substitute Andreas Skov Olsen and firing home with his right foot.
Any chance the Swiss had of mounting a comeback disappeared when captain Granit Xhaka, who was booked in the aftermath of the goal, was dismissed in the 87th minute after picking up his second yellow card.
Danish captain Pierre Emil Hojbjerg wrapped up the victory with a stoppage-time goal from another Skov Olsen assist to leave Denmark top of the group after a scoreless draw between Spain and Serbia.