Just as the match looked set for stalemate, Mehmet pounced in a crowded area to beat Tim Krul with the aid of a deflection.
It was cruel luck on the home side, but it again showed their capacity to leak late goals.
Pedro Moutinho saw an early shot deflected wide and Scott Arfield then had the ball stolen away from him by Garry Brady as he was poised to shoot.
Moutinho then pounced on a through ball from Darren Barr but his low cross was thumped for a corner by David van Zanten before it could do any serious damage.
Falkirk had an escape midway through the first half when goalkeeper Krul hit Stewart Kean's back with a clearance, but Falkirk managed to scramble the ball clear.
Arfield tried his luck from 20 yards but the ball struck Moutinho and bounced harmlessly wide.
Hugh Murray sent Mehmet clear but Barr managed to get his tackle in just as the striker was poised to have a shot.
There were half-hearted appeals for a free-kick at the edge of the area but Barr managed to get enough of the ball to satisfy referee Craig MacKay.
Krul then flapped at a cross but Jack Ross rescued him to clear the ball to safety with Saints looking threatening.
It took 34 minutes for Saints to have a shot at goal and even then Brady's 25-yard strike was wayward and Krul was not troubled.
Arfield swiped a shot well wide as half-time approached but neither goalkeeper was tested in the opening 45 minutes.
But Chris Smith had to react smartly to get down to a Michael Higdon shot within three minutes of the restart and then Mehmet blocked a Moutinho shot with his goalkeeper beaten.
Mehmet then set the ball up for Mark Corcoran at the other end and his shot clipped the crossbar with Falkirk all at sea.
But the response was swift. Barr looked to have scored when his downwards header beat Smith but Gary Mason blocked it on the line.
Murray should have scored after an hour when he was unmarked at the back post, but his diving header was saved by the alert Krul.
Smith then looked uncomfortable dealing with a looping cross from substitute Liam Craig and was glad to see the ball bounce off the top of his crossbar and drift wide.
But Mehmet made the difference in the final minute when he took a pass from Kean to deliver the killer blow.