Current world champion Wright was a doubt ahead of the tournament after suffering with gallstones. However, in true warrior fashion, the Scotsman battled through the pain and put on a classy performance to beat Joe Cullen 10-6.
Snakebite, who averaged 97, will take on Simon Whitlock in the fifth round of the event.
World number one Gerwyn Price also progressed following a 10-5 victory over an out-of-form Krzysztof Ratajski.
The Iceman, who has lost just two of his last 16 competitive games, will set his sights on England's Jamie Hughes in the next round.
Michael van Gerwen, who is enjoying a resurgence in recent weeks, is also through after beating Gary Anderson 10-7.
The Dutchman didn't quite hit his peak with an average of 93, but he did take out the current world number six which is likely to give Mighty Mike plenty of confidence going forward.
Elsewhere, James Wade will continue the defence of his title after beating Steve Lennon in what proved to be the match of the tournament so far.
Lennon, who averaged 105, was playing some of the best darts of his career, however his efforts proved fruitless as Wade, who enjoyed a 104 average, showed championship mentality to claim a 10-6 win.
Speaking of champions, two-time winner of the UK Open Raymond van Barneveld endured a third-round exit at the hands of Ally Pally nine-dart hero William Borland.
Barney averaged five points higher than the Scot, but failed to translate his scoring advantage into legs as Big Willie sealed a dramatic 6-5 victory.
Despite taking out the Dutchman, Borland was unable to extend his stay in Minehead into the weekend following a 10-6 loss to Dirk van Duijvenbode in the fourth round.
Adrian Lewis suffered similar fate. The two-time world champion averaged 108 as he demolished Germany's Fabian Schmutzler in his first match.
However, Jackpot couldn't book his place in the fifth round after falling to a 10-7 defeat against Ryan Searle.
Meanwhile, 2019 UK Open champion Nathan Aspinall bowed out of the tournament in the same round following a low-quality but highly dramatic 10-9 loss to Callan Rydz.
Last year's finalist Luke Humphries and former world champion Rob Cross also faced eliminations at the hands of Ian White and Gabriel Clemens.
World number eight Jonny Clayton almost followed suit, however the Welshman was able to keep his cool and prevail 10-9 in a tough battle with Martin Lukeman.