You probably don't need a regular expression for this:

if ( 0 == index $str1, $str2 )
{
    print "match\n";
}
else
{
    print "nomatch\n";
}
[download]

However, if you really want to use a regular expression then you need to escape the meta-characters in $str2:

if( $str1 =~ /^\Q$str2/ )
{
    print "match\n";
}
else
{
    print "nomatch\n";
}
[download]