The world no.21, with his left knee strapped, got off to a lacklustre start, looking uncomfortable at times.
His Ukrainian opponent took full advantage to breeze through and take the first set 6-1, but Kyrgios sprang into action in the second set.
Sealed with an ace - Kyrgios takes Brisbane title in impressive fashion
As with his second round match against Matthew Ebden, the Australian focussed his energy, breaking Dolgopolov and wrapping up the second set by 6-3 in 29 minutes.
Breaking in the seventh game of the third set, despite his sluggish start Kyrgios took the decider 6-4 to secure his semi-final place in 90 minutes.
Speaking after the match, Kyrgios said of his poor start: "As bad as it is to say, it's kind of like a tap. I can turn it on when I want to."
Kyrgios will play either Great Britain's Kyle Edmund or the top seed - and defending champion - of the tournament, Grigor Dimitrov, in the semi-finals.
There was further success for the host nation, elsewhere in the tournament on Friday, as Australia's Alex De Minaur won his quarter-final against Michael Mmoh 6-4, 6-0.
Meanwhile USA's Ryan Harrison got the better of Uzbekistan's Denis Istomin after he retired at 7-6, 4-2.
Edmund meets Dimitrov later on Friday.
Kyrgios gets Brisbane title boost ahead of Australian Open