For your limited example, sort already does exactly what you want. I guess though that what you want isn't what sort does for the following example:

firstcase_1
secondcase_4
secondcase_11
secondcase_012
[download]

What you want is called "natural sorting", and tye has a nice node on it - our categorized Questions and Answers also has How do I do a natural sort on an array?.

#!/usr/bin/perl
use strict;
use warnings;
use Data::Dumper 'Dumper';

my @files = qw(firstcase_1 firstcase_2 firstcase_10 secondcase_4 secon
+dcase_1 secondcase_11);

my @sorted = sort { $a =~ /^(\w+)_(\d+)$/; my ($a1,$a2) = ($1,$2);
                    $b =~ /^(\w+)_(\d+)$/; my ($b1,$b2) = ($1,$2);
                    $a1 cmp $b1 || $a2 <=> $b2 }
             @files;

print Dumper(\@sorted);
[download]

use Sort::Naturally;
@list = nsort(qw(
   foo12a foo12z foo13a foo 14 9x foo12 fooa foolio Foolio Foo12a
  ));
  print join(' ', @list), "\n";
[download]

or Sort::Key::Natural that is much faster...

use Sort::Key::Natural qw(natsort);

@sorted = natsort qw(foo12a foo12z foo13a foo 14 9x foo12 fooa foolio 
+Foolio Foo12a);

[download]