Falkirk had given the Edinburgh side a real battering in the opening 20 minutes, but a combination of bad luck and poor finishing meant they drew a blank.
Hibs central defence looked at sixes and sevens and Danny McBreen fired a good chance wide as early as 20 seconds into the match.
Then strike partner Darryl Duffy blazed over the bar from 15 yards before midfielder Steve Thomson also missed the target after Duffy had set him up.
And even when McBreen did hit the target after 24 minutes Hibs full-back David Murphy popped up on the line to head clear.
So it must have been hard to take when Brown put Hibs in front in the 36th minute.
He picked up a loose ball 40 yards out and drifted past Tiago Rodrigues and Thomson before drilling a low left-foot shot past the helpless Matt Glennon from 20 yards.
Falkirk almost replied immediately when Vitor Santos' shot was pushed around the post by Zbigniew Malkowski, but Hibs appeared to have weathered the storm and looked stronger for Brown's goal.
Dean Shiels was only a foot away after a twisting run in the box, and just after the interval Garry O'Connor missed a good chance to touch home Shiels' driven cross, before Glennon spread himself to save from the new Scotland star.
And despite Falkirk's bright start the game was over as a contest after 62 minutes when Shiels was tripped inside the box by Andy Lawrie allowing Derek Riordan to bag his first goal of the season from the penalty spot.
Hibs were able to coast home from this point although Malkowski did make fine late saves from Duffy and a deflected Vitor Santos drive.
But Falkirk are going to have to find a finisher if they want to stay clear of the end of season relegation struggle.