The problem is that you don't want to read the whole file into memory at once - yet the original file has to be modified. The following is a solution:

use strict;

my $match = 'var2 = 3';
my $in = 'in.txt';
my $tmp = 'temp.txt';
my $ext = 'extract.txt';

my ($inh, $tmph, $exth, $last);

open($inh, $in);
open($tmph, ">$tmp");
open($exth, ">$ext");

while (<$inh>) {
    if (index($_, $match) == -1) {
        print $tmph $last; $last = $_; next;
    }
    print $exth $last, $_; $last = '';
    print $exth (scalar<$inh>) for (1..9);
}

close($inh);
close($tmph);
close($exth);

unlink($in);
rename($tmp, $in);
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The modified version of the original file is written to a temp file, then the original file is deleted and the temp file is renamed. The script works - I tested it using your sample input data.