The Dubliner started well, a break of 88 giving him the opening frame and then taking a scrappy second frame.
But Bingham hit back to level at the interval.
Doherty regained the lead at 3-2 when, at 47-27 ahead, he produced a sensational snooker with two reds left on the table. The snooker cost Bingham 20 points and Doherty potted a red to make sure of the frame. A break of 52 saw Bingham level at 3-3 and he took the next two as well for the match as Doherty made a succession of errors.
Meanwhile, Pankaj Advani’s history-making run was halted in comprehensive fashion by world number two Judd Trump.
The 27-year-old from Bangalore became the first Indian player to make the quarter-finals of a ranking event when he beat Graeme Dott 4-1 on Thursday.
But he found Trump a different proposition in Newport, with the former UK Championship winner wrapping up a 5-2 victory.
Trump was on course for a whitewash when he went 4-0 ahead, but Advani showed some resolve to get back to 4-2, turning around a 59-1 deficit in the sixth frame.
However, Trump raised his game and quick-fire 69 in the next frame to advance.