my ($passwd_lit, $path_lit) = map text_to_shell_lit, $passwd, $path;
system("mysqldump --add-drop-table -uroot -p$passwd_lit mydatabase | g
+zip -9c > $path_lit");
[download]

my $passwd_lit = $passwd;
my $path_lit   = $path;
s#'#'\\''#g for $passwd_lit, $path_lit;
system("mysqldump --add-drop-table -uroot -p'$passwd_lit' mydatabase |
+ gzip -9c > '$path_lit'");
[download]

but to each his own.

By the way, your quoting was buggy. "\" is not special in sh single-quoted literals.