Brown said: "I have been here at Manchester United since I was a boy.
In addition to securing a place in Moscow, United are favourites to pip Chelsea and retain their crown.
Brown said: "If anyone had told us at the start of the season we would be in the final and needed to win two games to take the title, we would have been delighted.
"We are all happy to go through. Now we aim to do our best in Moscow.
"The lads are buzzing at the thought of being in the final."
United refused to buckle when Barcelona stepped up the tempo in search of a crucial goal and Brown said: "It was a great performance as we stopped them scoring.
"We dug in, knowing it was going to be difficult as they have some great players.
"They put a lot of pressure on us but we did brilliantly over the two legs and deserve to be in the final.
"We showed we can play as well as them. The fans were great as well. They got us going in the game."
Next up for United are West Ham at Old Trafford on Saturday and they complete the campaign at Wigan a week on Sunday.
Brown added: "They are two tough games but this victory against Barcelona will help us.
"We know if we play as well as we did in the Champions League then we have a great chance of retaining our title."