Falkirk took control from the start. Latapy was just inches away from giving them the lead after 13 minutes with a shot on the turn which ruffled the side-netting.
Thirty-seven year-old Latapy pulled the strings in midfield, and Livingston did not have a player who could match his creativity.
He gave Falkirk the lead after 28 minutes when he accepted a pass from Daniel McBreen at the edge of the area before firing a low shot past Roddy McKenzie.
Livingston toiled to get back into the game and had little to offer up front with Falkirk goalkeeper Matt Glennon not having a single shot to save in the first half.
McKenzie prevented Latapy from doubling Falkirk's lead 10 minutes from half-time when he leapt to his left to beat out the midfield player's shot.
The goalkeeper was caught out a couple of minutes later by a stunning Latapy free-kick, but the ball flashed just wide of his right-hand post.
It was one-way traffic and Vitor Lima was just a yard too high with a 25-yard effort after Latapy had cut the ball back into his path.
James McPake had a glorious chance to equalise nine minutes after half-time when he was through on Matt Glennon, but shot straight at the goalkeeper.
Glennon then did well to claw an overhead kick from McPake off his line as Livingston carried more threat.
But Falkirk went two up when Duffy scored with the aid of a deflection after the ball broke to him kindly off Livingston substitute Greg Strong. It was enough to kill off Livingston.