Colin Nish struck twice in the first half and defender Paul Hanlon scored what proved to be the winner with a close-in header a couple of minutes after the break.
Falkirk will feel they deserved a point but Hibs had their chances to finish off their opponents in style.
Hibs could hardly have wished for a better start. John Rankin slipped the ball inside to Nish, who had time to take a touch and then slip the ball inside Robert Olejnik at his near post.
Olejnik then had to throw himself full length to cut out a low cross from Dean Shiels after Nish had headed the ball into his path.
Falkirk tried to respond and Gerard Aafjes lashed a shot yards wide from good position at the edge of the area.
But Hibs were clearly in the mood. A superb flick from Nish sent Alan O'Brien scampering clear but Lee Bullen made the saving tackle.
Fletcher then laid the ball off to Nish and his strike partner should have done better than steer his eight-yard shot wide.
Olejnik then produced a reaction save to deny Fletcher after Barr had headed the ball into the striker's path.
O'Brien struck the side-netting after a flowing Hibs move ended with Fletcher setting up his team-mate who should have done better.
It was 2-0 after 25 minutes when a deep cross from Paul Hanlon found its way to the back post and the unmarked Nish stooped to head in an easy goal.
But Falkirk pulled a goal back five minutes later when Michael Higdon looped a header beyond Andy McNeil's reach from a pinpoint cross from Aafjes.
Rankin then had the miss of the match when O'Brien squared the ball and he somehow screwed his shot wide with only Olejnik to beat.
It was 3-1 after 47 minutes when Paul Hanlon headed in from a yard out after Nish had headed a Shiels' free-kick across goal.
But Higdon pulled a goal back seven minutes later when he pounced on a through ball from Jackie McNamara to pick his spot.
Fletcher struck the crossbar with a 25-yard shot which beat Olejnik and Hibs should have put the game out of sight.
Falkirk pressed for the final minutes in a bid to secure a point but Hibs held strong at the back.
Substitute Mark Stewart came closes to salvaging it two minutes from time when he struck the post from an acute angle.