Evans, fresh from a convincing victory against Juan Londero in the first round, only needed 75 minutes to secure a 7-5 6-2 win over Bublik and book his spot in the last eight.
Kazakhstan’s Bublik lost his serve to go down 5-4 in the first set but broke back for 5-5.
Evans, ranked 33 in the world, responded by winning a third break point to take a 6-5 lead and then served to love to take the opener.
The second set was a more pedestrian affair for the Briton, as he won five of the first six games and served out the match. Evans will now face either third seed Andrey Rublev or Sam Querrey for a place in the semi-finals in Australia.
Edmund had to work a little harder for his victory at the ASB Classic in New Zealand, where he was taken to three sets in a rain-affected clash before overcoming Alejandro Davidovich Fokina to record his first win of 2020.
The 25-year-old world number 69 won the first set in relatively-comfortable fashion but saw his opponent hit back in a second set that included a lengthy stoppage for rain when Davidovich Fokina was serving at 5-4 up.
When play resumed after a break of almost 90 minutes, the Spanish wild card duly wrapped up the set to send the match to a decider, where Edmund emerged triumphant to claim a 6-2 4-6 6-3 win.