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.
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.
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.
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.
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.
