Greig Laidlaw's six penalties kept Scotland in the game until the closing stages but the hosts were unable to close the gap, ensuring there was to be no third straight win in the RBS 6 Nations and their hopes of securing the title are now over.
While Laidlaw assumed the kicking duties from the tee, it was Weir calling the shots from fly-half, starting his first match for Scotland, having won three previous caps from the bench.
And Brown believes Weir can be more than satisfied with his performance.
"Duncan Weir should be pretty happy after his first start for Scotland," said Brown.
"He is a young lad, but very mature with it, and he is a very good reader of a game. He was brought in to play to his strengths, to kick the ball to the corners and keep us going forward, and I think we saw a lot of good stuff from him.
"But the game, for all of us as a group, is about doing the good things over and over again for 80 minutes.
"Duncan was not at all nervous beforehand. Nothing much fazes him. To see him like that in the dressing room beforehand was very reassuring for the rest of us, as fly-half is such an important position."