toniax: In response to your revised question, instead of your script to read a file and replace only the first line that matches your pattern I would use in-place editing, read and print each line and update a variable when the I've found and replaced the first occurrence of the pattern.

#!/usr/bin/perl
use strict;
use warnings;

my $filename = '/home/david/Programming/Perl/data/var.txt';

#So you can use diamond operator and inplace editing
push @ARGV, $filename;

#Enable inplace editing. Copy of input file saved as 'var.txt.bk'
$^I = ".bk"; 

my $replaced = 0;

while(<>){
    chomp;
    if ($replaced == 0
        and s/the line to be replaced/this is the new line/){
        $replaced++;
    }
    print "$_\n";
}
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