But John Hughes' side were indebted to goalkeeper Matt Glennon, who pulled off a string of unbelievable saves.
Falkirk played the final stages with only 10n men after skipper Craig Ireland was sent off when he fouled Bartosz Tarachulski when he was the last defender in the 66th minute.
Dunfermline almost scored after only 90 seconds when Gary Mason cut the ball back for Bartosz Tarachulski to thunder towards goal, but Glennon produced an acrobatic leap to push the ball for a corner.
Liam Horsted was next to go close after being sent clear by Yannick Zambernard,i but Glennon blocked his angled shot.
But Falkirk came back strongly and Allan McGregor leapt to his left to push out a 20-yard shot from Alan Gow.
Falkirk made the breakthrough five minutes from the break with a simply-constructed goal. Russell Latapy fed the ball to Duffy, who turned Scott Thomson at the edge of the box before drilling the ball past McGregor.
Falkirk took heart from the goal and could have added a second a minute later when Steve Thomson wriggled clear, but he pulled his shot across the face of the goal.
After Ireland's sending off, it was backs to the wall for Falkirk. Thomson thought he had scored on three separate occasions in the final moments only to see Glennon pull off heroics – pushing one header against a post.