A largely-scrappy contest saw Cope take three of his five frames on the final two balls, and the last four in a row after a 49 break from Selby had levelled the match at 1-1.
Selby had clocked up runs worth 33, 43 and 21 in those scrappy frames, the highest breaks of each, only to lose to Cope on the final colour trail.
The 27-year-old Cope then racked up a 71 break, the highest of the match, to seal victory in the sixth frame.
Either Stuart Bingham or Tom Ford will serve as Cope's second-round opponent when the last-16 stage gets underway on Thursday, with Bingham and Ford in action on Wednesday morning.
Also in the day's opening session, Stephen Maguire advanced at the expense of Peter Ebdon in a mammoth opening round match which the Scot won 5-3.
2002 World champion Ebdon twice took the lead only to be pegged back by breaks of 78 and 49.
An Ebdon 69 and a Maguire 73 kept the two level at three frames apiece before Maguire seized control with a run to 58 before fighting off Ebdon's attempts at salvaging frame seven with snookers.
And with the Englishman needing frame eight to stay alive, the notorious grinder put up a stubborn resistance and capitalised on several unforced errors by Maguire in his efforts to force a decider.
But after over an hour of play in the frame, Ebdon finally saw his chances of winning the necessary snookers disappear and Maguire will now meet Ali Carter who won a final-frame play-off against Robert Milkins.
The final two wild card qualifiers for a spot in round one were also completed, with Chen Feilong seeing off Fergal O'Brien 5-2 to secure a Wednesday fixture against 2010 Crucible champion Neil Robertson.
Compatriot Jin Long, meanwhile, defeated Jimmy Robertson 5-4 in the longest match of the session to set up a meeting with 2006's World Championship winner Graeme Dott.
- Stephen Maguire
- Peter Ebdon
- Stuart Bingham