The 27-year-old seemed to deal with the pressure superbly as he brushed off a silly foul on a red to take the first frame comfortably.
The second frame was an attritional affair as no balls were potted for 15 minutes, but Cope pulled out a fantastic nerve-free frame of his own to level proceedings.
Walden hit back in some style, falling just two points short of a century break to make it 2-1 before coming from 40 points down to clinch the match in the fourth.
Walden's victory served as a delightful preamble to the afternoon's main event, at least from the audience's perspective, as Ronnie O'Sullivan defeated Stephen Hendry 3-1.
Billed as a clash between snooker's two biggest heavyweights, the quality did not disappoint as both players produced some fantastic snooker at times, with very few mistakes.
The first two frames were a question of who was going to add to their career century totals, with O'Sullivan's 605 only second in the all-time list to Hendry's staggering 748.
In the end, neither player managed another hundred as the Rocket took the first frame with 93 before Hendry hit back with 95.
O'Sullivan went on to edge the third frame before a fabulous break of the pack in the fourth provided him with opportunity to see it home, which he duly did, despite a miss on a routine red.
Stephen Maguire did not have to do much to see off an out-of-sorts Alan McManus 3-0 in the final match of the afternoon's play.
Labelled as a derby, the two Scottish players unsurprisingly failed to match the quality shown in the previous two matches, with McManus showing some horrific form throughout.
Martin Gould did his best Stephen 'The Ice Man' Hendry impression to coolly see off Stephen Lee 3-0 in the final match of the last 16.
Just as he did against Matthew Couch, the Londoner never lost his focus as he rattled off the three frames in an hour, handling the pressure brilliantly and firing in some super shots.
All three frames followed a similar pattern, as Gould accrued enough points without hitting any bug breaks, although his expert clearance to take the second will have given his quarter-final opponent Peter Ebdon food for thought.
Maguire chipped away in each of the first two frames without making any high breaks, but still eased to a 2-0 lead.
McManus disappointingly offered nothing to prevent a whitewash, with a decent half-century break of 77 from Maguire seeing things off.
World champion Neil Robertson played some unbelievable snooker to knock out a resilient Andrew Higginson 3-2 in the match of the tournament so far.
Higginson didn't do a lot wrong in the first frame - and, in truth, the entire match - but he couldn't stop the Australian from going into a 1-0 lead thanks to some devastating potting.
The Englishman responded fantastically though as a canny break of 46 saw him level proceedings in the second, before he went 60-11 ahead in the third and looked like he would go in front.
But there was no stopping Robertson in this sort of form as he cleared the table magnificently to retake his lead to rapturous applause.
Once again though Higginson somehow found the resolve to win the fourth and take the match into sudden death, which became a very cagey affair that Robertson ultimately edged to set up a tasty quarter-final match with Ricky Walden.