The Hoops beat Braga 2-1 at home in the second leg of their third qualifying round clash but lost 4-2 on aggregate and will be now in the draw for Europe's second-tier competition.
Brown insists there is no reason why Celtic cannot emulate the efforts of the Cottagers last season, when they reached the Europa League final before losing to Atletico Madrid.
"We just need to go on a run and see how far we can go," the Scotland midfielder told the Scottish Sun.
"We've also got to keep the home record as good as possible.
"When you see Fulham getting all the way to the final like that it can inspire you. And look at Rangers two years before that.
"Anything can happen and that's what we're looking to do now."
Brown rejected the idea that the Champions League qualifiers came too soon for Neil Lennon's side, which is still in the throes of refurbishment.
He said: "I don't think the ties came too early for us. We just didn't do ourselves justice out in Braga.
"But we came back here and we were hoping to get an early goal. Unfortunately it happened the other way around.
"The Europa League is a consolation. But we wanted to play Champions League football, especially for the fans, and this club deserves to be in it every season.
"I don't think the Europa League is a more realistic level for us.
"We showed that we can beat Braga and at home I think we can beat anybody on our day.
"We've shown that in the past and we'll show it in the future as well."