Britain assumed the early lead in the dramatic tie thanks to a dominant performance from Kyle Edmund, but were pegged back when Bublik overcame Dan Evans in three sets.
An early exit would have awaited had Murray and Skupski lost out in the decisive rubber, but they rose to the occasion and completed an emphatic win.
GB forced the breakthrough early on with a break to lead 3-1 and then 5-1 before cruising to the opening set in style.
The second set followed a similar pattern as Murray and Skupski raced into a quick lead with a break in the first game, and a further assault on the Kazakh service wrapped up the match in comprehensive fashion.
Andy Murray and his team-mates watched on nervously as the match began, but it was clear early on that the British pair were the class act on the court.
Britain will now take on Germany, led by Philipp Kohlschreiber and Jan-Lennard Struff, in the last eight with the tie taking place on Friday afternoon.
Andy Murray and Neal Skupski of Great Britain celebrate following the Great Britain v Kazakhstan doubles match during Day 4 of the 2019 Davis Cup at La Caja Magica