In the third qualifying round first-leg in Poland last week, Ronny Deila's men suffered a nightmare result which left their hopes of progressing to the play-off round hanging by a thread.
Celtic crumbled after taking the lead, conceding four times and having Efe Ambrose sent off, and it could have been much worse for the Scottish champions had Ivica Vrdoljak not missed two penalties for the hosts.
Brown, currently out injured with a torn hamstring, hopes the Celtic faithful will flock in their numbers to Murrayfield to help the team produce a remarkable fightback.
"Wednesday night is going to be a huge game at Murrayfield," he told the club's official website.
"The fans, I'm sure, will come out again in their thousands and that's definitely going to help us.
"It's going to be hard because we need three goals and we need to show them how good we are. But I believe that we can do it.
"Everyone knows it was a disappointing result and it was hard to take, but we've got a chance here to go and put it right and I'm sure we will."
Last season, Celtic lost 2-0 in Kazakhstan to Shakhter Karagandy, only to turn the tie around with a 3-0 success back on home soil, and Brown is confident of a repeat performance this time around.
"The Shakhter game last season was huge for us," he said. "We scored three goals and we're going to need to do it again this week.
"The fans were fantastic in the game against Reykjavik [in the previous round], and I'm sure they'll create another brilliant atmosphere at Murrayfield on Wednesday.
"Our fans are so important to us, and we've seen how they can help put pressure on the opposition, which is what we'll be looking to do against Legia Warsaw."