Mehmedi's cracker prevented Romania from adding to their solitary victory at the European Championships, which came against England 16 years ago.
It leaves them with just one point from two matches, although with their final game to come against minnows Albania, they will still have high hopes of qualifying for the knock-out stages.
Switzerland, who won their first Group A game against the Albanians, almost took an early lead Haris Seferovic turned away from Vlad Chiriches in the penalty area and curled his shot inches wide.
Xherdan Shaqiri then put Seferovic in on goal again but his low shot was kept out by the feet of keeper Ciprian Tatarusanu.
But Romania were gifted the lead after 18 minutes when Stephan Lichtsteiner pulled Alexandru Chipciu's shirt in the area and Russian referee Sergei Karasev pointed to the spot.
Stancu, who scored a penalty in the opening night defeat against France, made it two from two when he sent Swiss keeper Yann Sommer the wrong way.
Switzerland came close to hitting back in spectacular fashion when Fabian Schar hit a dipping strike which Tatarusanu had to tip over the crossbar.
Romania almost doubled their lead when Switzerland failed to clear a high ball into the box, but when it fell to Cristian Sapunaru he stabbed his shot against a post and wide from eight yards.
Moments after the break Romanian half-time sub Ovidiu Hoban dragged a shot across goal before some fine defending from Johan Djourou in the six-yard box cleared Gabriel Torje's dangerous low cross.
Instead Switzerland hauled themselves level just before the hour mark when a half-cleared corner dropped to Mehmedi, who lashed his volley past Tatarusanu and into the net.
Romania tried to hit back and Torje's audacious attempt from 40 yards, with a free-kick from which everyone expected a cross, caused Sommer an anxious moment as he scrambled over to push the ball away.
But as Romania visibly tired Switzerland threatened a winner, but Shaqiri should have done better with a cross in the area which he had time to chest down, but instead launched himself into an overhead kick and scuffed the ball wide.
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