Selby had taken a 2-0 lead in their round three match in Yushan, China, but the three-time world champion barely got a sniff after that.
Bingham, the 2015 world champion, reeled off five frames in a row to reach the quarter-finals.
‘Magical snooker’ – Bingham roars back to crush Selby
Northern Ireland Open
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Breaks of 108, 97, 114 and 104 took Bingham to the brink, and a sensational double helped seal a 117 in the final frame.
Bingham will face John Higgins in the last eight after the Scot secured a final-frame victory over compatriot Graeme Dott.
Dott got off to a flying start with breaks of 104 and 80, but a 71 and 99 saw Higgins level the match.
Higgins then trailed 3-2 and 4-3, but rallied to defy Dott and win the last two frames with a 60 and 69.
Elsewhere, Thepchaiya Un-Nooh prevailed in his all-Thai clash with Sunny Akani.
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Un-Nooh bounced back from losing the first frame to lead 3-1, only for Akani to level at 3-3.
However, breaks of 62 and 74 helped Un-Nooh get over the line and set up a tie against David Gilbert.
Gilbert overcame Mark Allen 5-3, winning four of the last five frames to reach the quarters.
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