It looked like coach Stephen Larkham's gamble of leaving key Wallabies at home wasn't going to pay off with the Brumbies trailing until the 74th minute when Reimer rumbled over for his second.
Making their first trip to Fiji, the Brumbies became the first Australian side to win on the Pacific island, with the Drua's record at home now two losses from their past 11 matches.
Larkham left Test prop James Slipper, lock Nick Frost, fullback Tom Wright and centre Len Ikitau in Canberra, while other Wallabies stars including Rob Valetini were out through injury.
The risk looked a smart move early on, with the visitors storming to a 17-7 lead, helped by two Drua yellow cards.
One of those was against Vuate Karawalevu for a deliberate knock-down, but the winger made amends in the second half, scoring twice to give his team the winning edge.
The Brumbies scored three first-half tries, through hooker Billy Pollard, centre Ollie Sapsford and scrumhalf Ryan Lonergan, who darted down the short-side off a line-out.
But 12 handling errors to the home side's two and Noah Lolesio missing two conversions brought their good work undone, with a converted try by centre Ponipate Loganimasi in the 37th minute giving the Drua an 18-17 halftime advantage.
The lead continued to change hands, with debutant Karawalevu's double putting his team up 32-24 at the 60 minute mark.
Forced off early in the second half for an HIA, Reimer then used the Brumbies' dominant set-piece to put his team back in the match.
As well as contributing four turnovers, Reimer dived across the line in the 67th minute and burrowed over again in the 74th for the visitors to go ahead 34-32.
Replacement five-eighth Declan Meredith added the extras for a four-point win.
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