For 55 minutes at Wembley, the Black Cats threatened to pull off a major shock as they led Manchester City 1-0 having enjoyed the better of the first half.
But stunning finishes from Yaya Toure and Samir Nasri inside two second-half minutes handed City the advantage, and they eventually ran out 3-1 winners.
Sunderland could earn themselves another trip to Wembley next weekend when they face Hull in the FA Cup quarter-finals, but securing their Barclays Premier League status will be the main focus.
Asked what manager Gus Poyet had said in the dressing room after the final whistle, 34-year-old Brown said: "He just said, 'Listen, we have got important games coming up now and if we can play like that in the first half, we can play like that every week'.
"We have just got to take that in and make sure we do it in the league now.
"They scored two fantastic goals and managed to kill us with a third, so it was disappointing.
"But the lads can keep their heads up high and we have to start taking that sort of form we had in the first half into the league now."
Poyet will hope a display full of grit, character and no little skill will provide the springboard for a late-season surge, and Brown sees no reason what that cannot be the case.
He said: "Yes, why not? We can do it against the big teams, so we have just got to keep it going against the teams around us now."
For all Sunderland's best efforts, City's quality eventually told and there was little Brown and his fellow defenders or goalkeeper Vito Mannone could do to keep the opposition at bay.
Toure's imperious 55th-minute equaliser was swept into the top corner in stunning fashion to spark a powerful fightback.
Brown said: "When it's a final, I always think it's going to take top-quality goals to win finals, and they were.
"Even when he [Toure] was shooting, I was thinking we were okay, and he managed to put it in the top bin.
"It's just the way it goes. In that sort of sense, we are disappointed, but they were fantastic goals to get them into the lead.
"Considering they scored three goals, they never really had a clear-cut chance in the sense that they got behind us or battered us.
"But that's football, they have got quality players and that's what quality players do.
"At times, we had joy against them especially in the first half, but in the second half you could see we tired a little bit and they were just throwing all the pressure on us, and they scored the three goals."
The Black Cats headed back to Wearside wondering what might have been having given their illustrious opponents a major fright, and while they were rightly proud of their efforts, there was genuine disappointment at the outcome.
Brown said: "We have done well to get here - we were never the favourites to get here and we can always take that for as long as we carry on.
"But it's a final at the end of the day, there are two teams on the pitch and considering we took the lead, it's a little bit disappointing that we couldn't score the second goal and make it even more difficult."