Here is a one-at-a-time Fisher-Yates shuffle but using an array of indices instead of modifying your array of numbers directly (if even avoids having to move around half of the extra array via splitsplice):

#!/usr/bin/perl -lw
use strict;

my @inviolable= getNumbers();
my $pickNext= genPickWNoReplace( \@inviolable );
while(  1  ) {
    print $pickNext->();
}

sub genPickWNoReplace
{
    my( $av )= @_;
    my @idx= 0..$#$av;
    return sub {
        die "No more items left to pick from"
            if  ! @idx;
        my $o= int rand @idx;
        my $pick= $av->[ $idx[$o] ];
        $idx[$o]= $idx[-1];
        pop @idx;
        return $pick;
    };
}
[download]

(update above thanks to fenLisesi's correction)