inet_aton returns the address as a 4 byte string. Each byte of the string represents one of the numbers of the IP address. In this format, they can be sorted lexically. 
use Socket qw( inet_aton );
use List::MoreUtils qw( apply );

my $file_name = "ipsvisiting.txt";

open (my $fh, '<', $file_name)
   or die("Unable to open file \"$file_name\": $!\n");

my @sorted = map { substr($_, 4) }
             sort
             map { inet_aton($_) . $_ }
             apply { chomp }
             <$fh>;

print "$_\n"
   foreach @sorted;

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I started writing this when there were no replies, but I got sidetracked. Hopefully, this approach hasn't been suggested already.

I made some change to your code:

In reply to Re: How to sort IP addresses by ikegami
in thread How to sort IP addresses by Anonymous Monk

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