I have made a few changes to your script:

• Added use warnings;
• Stored the regex in a variable, $re, to avoid duplication.
• Removed the unnecessary capturing parentheses from the regex; since you are only using $1, you only need 1 set.
• Check $first_number instead of $1. Same for $2.

Here is the code:

use strict;
use warnings;
use File::Copy;

sub files_sort {
    my $first_number; 
    my $second_number; 
    my $re = qr/(\d+)_\d+_duration_\w+_comptage_\d+.txt$/;
    if ($a =~ $re) {
        $first_number = $1; 
    }
    if ($b =~ $re) {
        $second_number = $1; 
    }
    if ($first_number < $second_number) {
        -1
    }
    elsif ($first_number > $second_number) {
        1
    }
    else {
        0
    }
}

my @TXT = glob("*.txt");
@TXT = sort files_sort @TXT;
print "$_\n" for @TXT;
[download]

Here is the output:

% ls -1 *.txt
A3_output_ZGZ22_03_20061022_duration_200mn_comptage_60.txt
M3_output_ZGZ22_02_20061022_duration_200mn_comptage_60.txt
M3_output_ZGZ22_12_20051022_duration_200mn_comptage_60.txt
M3_output_ZGZ22_12_20061022_duration_200mn_comptage_60.txt
% 689020.pl
M3_output_ZGZ22_02_20061022_duration_200mn_comptage_60.txt
A3_output_ZGZ22_03_20061022_duration_200mn_comptage_60.txt
M3_output_ZGZ22_12_20051022_duration_200mn_comptage_60.txt
M3_output_ZGZ22_12_20061022_duration_200mn_comptage_60.txt
[download]

Is this what you expect?

Caveat: you must make sure that all the *.txt files in your directory match your regex.