It took Falkirk until April last season to register a home league win but Barr's 10th-minute header was enough to win this one.
He met Russell Latapy's corner perfectly to bullet a header past Roddy McKenzie from six yards.
The home side went on to dominate the first half and would have led by a more handsome margin had their shooting not been erratic.
There was a let-off for the home side ten minutes from the break when Scott Higgins failed to cut out a Steve Simmons' cross but he recovered in time to knock the ball away from Freddie Daquin.
Alan Gow then shot straight at McKenze as the Dunfermline defence opened up invitingly again.
He had a more glaring miss moments shortly afterwards. Latapy laid the ball on a plate for him, but his left-foot finish was too high when he had time and space to pick his spot.
But Falkirk had to rely on Higgins to keep them ahead at the break. Simmons carved out a chance for himself with some clever footwork but Higgins was off his line quickly to somehow get a hand to his ten-yard shot.
The lively Latapy forced a diving save from McKenzie early in the second half before Dunfermline hit back.
Burchill was guilty of squandering another chance when he headed over from just four yards with Higgins exposed.
Gow almost scored a second for Falkirk at the death but his low shot was well saved by McKenzie.