Rose, the youngest player in the Canadian squad, got the opening goal for Canada after 25 minutes when she was on hand to side-foot home a low cross from Ashley Lawrence, after the defender had run half the length of the field on a quick counter-attack.
She then turned goal-maker seven minutes into the second half, when she crossed for Christine Sinclair to make it 2-0 from close in.
Brazil, who won silver in 2004 and 2008, looked tired after a punishing semi-final defeat against Sweden after extra time and penalties in the heat of Rio de Janeiro just three days ago.
The host nation pushed forward as the game went on and Beatriz pulled a goal back 11 minutes from time when she controlled the ball nicely in a packed box and turned to fire low into the net.
But the Canadians were worthy of their win and the scoreline could even have been wider, with the visitors smacking the woodwork once in each half.
Germany will play Sweden later on Saturday for the gold medal.
