Niall McGinn scored twice, taking his tally to 11 for the season, as the Dons defied the elements to climb above Saints and within three points of Wednesday's Boxing Day opponents Motherwell.
"Without question we were the better team," Brown said. "If it had been 0-0, which was possible until McGinn took the free-kick, I'd still have said we were the better team. We got the goals which gave us the points."
Brown selected 10 players eligible at under-21 level in his squad, seven more than SPL regulations require, as a result of eight injuries and one suspension, for goalkeeper Jamie Langfield.
Influential forward Jonny Hayes (knock) and Andrew Considine (ill) were the latest to miss out, but could return at Fir Park, should Brown opt to tinker with his side.
"We're hoping we might get a couple of players fit," Brown added.
"It will be hard to disturb that team because they played particularly well. If we think we can improve it we'll improve it."
Jason Brown made his first start of the season in goal after Langfield was suspended for his dismissal against Kilmarnock, but the Dons boss was tight-lipped over who would play at Motherwell.
Wales international Brown was limping throughout and his fitness could be a factor in the decision.
Brown said: "He was struggling before the game. We were worried about whether he could play because he got an injury in training on Thursday, his ankle. "We had Danny Rogers, the young goalie, ready."