I agree that this var formattedString = string.replace(/'/g, "\\'");
works very well, but since I used this part of code in PHP with the framework Prado (you can register the js script in a PHP class) I needed this sample working inside double quotes.
The solution that worked for me is that you need to put three \
and escape the double quotes.
"var string = \"l'avancement\";
var formattedString = string.replace(/'/g, \"\\\'\");"
I answer that question since I had trouble finding that three \
was the work around.