use strict; 
use warnings;

my $str = 'A string with a "double quoted" string in it';
(my $quoted = $str) =~ s/\"/\\\"/g;

print "$str\n$quoted\n";
[download]

Prints:

A string with a "double quoted" string in it
A string with a \"double quoted\" string in it
[download]

Where's the problem?