The Power, who had been adamant they wouldn't bring forward plans for Josh Carr to succeed under-fire coach Ken Hinkley at season's end, fought through a Sam Taylor defensive masterclass to prevail 9.12 (66) to 7.8 (50).
The Giants will sweat on Toby Greene's fitness ahead of next Saturday's clash with Brisbane at the Gabba after their skipper was substituted at halftime with a nasty corked left glute at Canberra's Manuka Oval.
With the Power needing a change of fortunes, and the heat on Hinkley, Rozee and Butters stood tall.
Butters (32 disposals) had a career-high 12 clearances, plus 20 contested possessions, while Rozee had 690 metres gained from 28 disposals.
Kane Farrell, Esava Ratugolea and Miles Bergman worked hard behind the ball in a gritty performance.
While Port relished their first win in Canberra since 2014, GWS coach Adam Kingsley will have plenty to ponder after the Giants (7-5) were held to just one goal in the second half.
Late in the first quarter, Greene copped a knee from Aliir Aliir to his left glute in a marking contest and hit the deck in discomfort.
Greene played on under some duress but had limited movement, and the Giants substituted him for Brent Daniels at halftime.
Taylor (15 intercept possessions, nine marks) worked over Port spearhead Mitch Georgiades and set the tone with support from partner-in-crime Jack Buckley (12 marks, 24 disposals).
Canberra local Tom Green (31 disposals) gave the Giants punch and power through the midfield, while Lachie Ash racked up plenty of the ball off half-back, and Jesse Hogan and Jake Riccardi kicked two goals apiece.
Willie Rioli (two goals) and Rozee nabbed the first two goals, but the Giants settled into the game and conditions in Canberra.
Port led by two at quarter-time before GWS took a three-point lead into the main break.
Port were arguably the better team from halftime onwards but were thwarted by their inefficiency and a resolute Taylor.
The third term was just the Giants' third scoreless quarter in the past 10 years, but Port led by just seven at the final change.
Hogan cut the deficit to a point early in the final term.
Substitute Cristian Moraes kicked his first AFL goal to extend Port's lead to seven, before Ollie Wines added another from a free kick, then Joe Richards dealt the knockout blow.
The Power (4-7) will head into next Sunday's clash with Melbourne with renewed confidence.
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