Another way to do it, while certainly not the most efficient, is to replace undefelements with a value guaranteed to both be the maximum value in the comparison and uniquely distinct from the original value set.

I show this inefficient technique more to sow seeds for inspiration for future problems you may run into where setting a maximum value might be a useful idea. Data transmogrification still has value in some cases; having that tool in your toolshed is useful.

Coupled with the approach used by LanX, you could probably craft a fairly efficient and reliable version of this approach.

In this example, I convert the original array such that undefelements are replaced with a unique identifier guaranteed to sort to the bottom of the list (based on the assumption this is a string comparison); the replacement string is guaranteed to not have been found in the original array (again, via an inefficient technique -- I force it to be longer than the longest element in the original array).

From there, the rest of the operation should be relatively intuitive.

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

# Constants (Operationally, if not functionally)
my $MAXCHR = chr 255;
my $HORRUL = '--------------------------------------------------------
+-----------------------';

# Globals
my $Maxval = '';
my @Unsorted = ( 'Dog', 'Cat', 'Bird', undef, 'Elephant', undef, 'Liza
+rd' );
dumpArray("Original:", @Unsorted);

# Build a maximum-value scalar:  Length = longest + 1
# We will temporarily replace all undefined elements with this value
# Since it is composed of all chr 255 values, it is the highest for so
+rt
# Since it is one character longer than the longest element, it unique
+ly 
#   identifies the element as a former undef
# Then sort can do its thing and we can detect these to convert back t
+o undef
my $maxlen = longestElement(@Unsorted);
while (length($Maxval) <= $maxlen) {
    $Maxval .= $MAXCHR;
}

# Build a working copy of the array with undef replaced by $Maxval
my @unsortedWork = ();
foreach (@Unsorted) {
    if (defined $_) {
        push @unsortedWork, $_;
    } else {
        push @unsortedWork, $Maxval;
    }
}

# Sort, Convert modified entries back to undef, and show results
my @sorted_custom = sort @unsortedWork;
my @sorted_final = ();
foreach (@sorted_custom) {
    if ($_ eq $Maxval) {
        push @sorted_final, undef;
    } else {
        push @sorted_final, $_;
    }
}
dumpArray("Custom Sort:", @sorted_final);

exit;

# Display an array with a title
sub dumpArray {
    my ($dumpTitle, @dumpValues) = @_;
    print "$HORRUL\n$dumpTitle\n$HORRUL\n";
    foreach my $dumpValue (@dumpValues) {
        if (defined $dumpValue) {
            print "$dumpValue\n";
        } else {
            print "(undef)\n";
        }
    }
    print "$HORRUL\n";
}

sub longestElement {
    my $maxlen = 0;
    foreach(@_) {
        if (defined $_) {
            if (length > $maxlen) { $maxlen = length; }
        }
    }
    return $maxlen;
}
[download]

Results:

S:\Steve\Dev\PerlMonks\P-2017-06-12@0734-sort-undef>sort1.pl
----------------------------------------------------------------------
+---------
Original:
----------------------------------------------------------------------
+---------
Dog
Cat
Bird
(undef)
Elephant
(undef)
Lizard
----------------------------------------------------------------------
+---------
----------------------------------------------------------------------
+---------
Custom Sort:
----------------------------------------------------------------------
+---------
Bird
Cat
Dog
Elephant
Lizard
(undef)
(undef)
----------------------------------------------------------------------
+---------

S:\Steve\Dev\PerlMonks\P-2017-06-12@0734-sort-undef>
[download]