Having seen his side soak up plenty of pressure, the Scotland skipper got the crucial goal a minute after the hour mark.
After a series of short passes appeared to have to led to nothing, Brown seized on a loose ball and struck home a fierce low effort to secure the win.
Scotland's unbeaten run now stands at four matches and they have won three of their last four away games.
Norway made three changes from their 2-1 defeat to Denmark on Friday, handing goalkeeper Orjan Nyland his international debut while Magnus Wolff Eikrem and Morten Gamst Pedersen also started.
Scotland manager Strachan had seen a host of players withdraw from the squad - Steven Fletcher, Charlie Mulgrew, Grant Hanley, Ross McCormack and Matt Gilks.
Derby County's Craig Bryson was handed his second cap and there were also starting berths for Russell Martin, Charlie Adam, Ikechi Anya and Steven Naismith.
Kris Boyd, whose most recent appearance for Scotland had come in 2010, had to be content with a place on the bench.
Nyland was nearly caught out early on when his poor clearance from outside the area found Anya, who controlled neatly before volleying just wide of an open goal.
Despite Scotland posing the first threat, the hosts were generally the brighter, with David Marshall forced into a sliding clearance with Morten Gamst Pedersen through on goal before saving an Ola Kamara header.
Norway continued to apply the pressure and Marcus Pedersen came close to turning home his namesake's delivery from the left.
A resolute Scottish backline held firm, although Eikrem managed to angle an effort just wide of Marshall's right-hand post two minutes before the break.
Eikrem was first to threaten in the second period, shooting straight at Marshall from the edge of the area.
Marcus Pedersen should have put Norway ahead two minutes later, but Marshall saved well with his legs from close range as the hosts' domination continued.
Substitute Mohammed Abdellaoue was the next to waste a presentable opportunity, heading into the ground and wide from a left-wing delivery.
Brown then punished the opposition for their profligacy, firing into the far corner of Nyland's net and extending Norway’s run of matches without a clean sheet to five.
Norway went in search of an equaliser, with Anders Konradsen having the pick of the chances only to put his effort wide from close range as Scotland held on to shut their opponents out.
