Led by Zak Butters and Connor Rozee, the Power, who had been adamant they wouldn't bring forward plans for Josh Carr to succeed Hinkley at season's end, prevailed 9.12 (66) to 7.8 (50) in Canberra on Saturday night.
The win over GWS, who lost skipper Toby Greene to a corked glute at half-time, moves Port Adelaide to 5-7 and just eight points outside the finals places.
When asked if the win eased the pressure, Hinkley said: "What pressure?"
When pressed, he said: "(It will) continue on, I suppose. It's just gonna continue on for the remainder of the season.
"But what I do know tonight is that helps us get back on track where we want to be.
"Clearly we're a long way off where we thought we would be and need to be.
"But at least we feel like we maybe snuck our head around the corner just a little bit just to have a look and see what's next and we've got a big game next week back at Adelaide."
When asked if finals were still Port's dream, Hinkley said: "Yeah, well why wouldn't they be?"
The Power will head into next Sunday's clash with Melbourne at Adelaide Oval with renewed confidence and Hinkley made it clear he believes finals are still on the agenda.
While Port relished their first win in Canberra since 2014, GWS coach Adam Kingsley will have plenty to ponder after the Giants (7-6) were held to just one goal in the second half.
"We expected what they did," Kingsley said.
"We planned really well for it. We just didn't do it as well as what they did.
"He (Hinkley) prepares his team and his players really well.
"It's clear that they love him and they play a strong brand of footy in response to when he's under pressure. That was evident tonight."
The Giants will sweat on Greene's fitness ahead of next Saturday's clash with Brisbane at the Gabba after he copped a knee from Aliir Aliir to his left glute/hip in a marking contest late in the first quarter.
Greene played on under some duress but had limited movement, and the Giants substituted him for Brent Daniels at halftime.
Butters (32 disposals) had a career-high 12 clearances, plus 20 contested possessions, while Rozee had 690 metres gained from 28 disposals as the pair shook off GWS tagger Toby Bedford.
Kane Farrell, Esava Ratugolea and Miles Bergman worked hard behind the ball.
GWS key back Sam Taylor (15 intercept possessions, nine marks) worked over Port spearhead Mitch Georgiades and turned on a defensive masterclass with support from partner-in-crime Jack Buckley (12 marks, 24 disposals).
Canberra local Tom Green (31 disposals, 11 clearances) gave the Giants punch and power through the midfield, while Jesse Hogan and Jake Riccardi kicked two goals apiece.
The game proved an arm wrestle until Port slammed home the final three goals of the fourth term to finish the job.
