A mix-up between skipper Rob Jones and goalkeeper Yves Makaba-Makalamby presented Mehmet with the ball outside the area and he kept his nerve to roll the ball in off a post.
Hibs' day was not helped when substitute Dean Shiels was sent off in the second half after reacting angrily to a challenge from Franco Miranda.
Lewis Stevenson had Hibs' first attempt at goal after only a couple of minutes but his shot sailed high over the top.
But Saints responded and Stewart Kean just failed to connect with a deep cross from Mehmet with David Murphy doing enough to put him off.
Hibs had a shout for a penalty when Mickael Antoine-Curier claimed he was impeded in the area by Andy Millen but referee Chris Boyle saw nothing untoward.
But the home side gifted St Mirren a goal after 13 minutes. Jones was too strong with a headed pass back and the ball cannoned off Makaba-Makalamby's legs to Mehmet, who kept his cool from outside the area to slot the ball in off the post.
Miranda displayed Saints' growing confidence when he tried to catch Makaba-Makalamby off guard as he looped in a 30-yard free-kick but the ball dipped onto the roof of the net.
A clever dummy from Mehmet created a shooting chance for Kean but he could not get hold of it properly and Makaba-Makalamby saved comfortably.
St Mirren were working hard all over the park and closing down opponents and not giving Hibs time to settle on the ball.
There was a glimmer of hope for Hibs after 30 minutes when Chris Smith failed to clutch a Merouane Zemmama corner and it broke to Jones, but the goalkeeper redeemed himself by reacting swiftly to block the defender's shot.
Millen then did superbly to get his head to a Zemmama cross to divert it away from Antoine-Curier who would have had a simple chance to net.
Stewart Kean fired just over late in the first half and Hibs changed it at half-time by bringing on Alan O'Brien and Shiels.
It gave them more urgency and Smith saved well from Stevenson before the midfielder had another decent effort which was just too high.
But Hibs' momentum was lost when Shiels was sent off in 64 minutes - much to the fury of the home support who felt the referee had over-reacted.
Brian Kerr almost equalised for Hibs after Smith had dropped a David Murphy corner but hooked the ball over.