if(defined $cache[$num]) {return $cache[$num];}
if($num < 0) {return 0;}
[download]

In Perl, -1 is a valid array index, meaning the last element of the array (-2 is the second-to-last, and so on). Move the if($num < 0) check before the if(defined $cache[$num]) check and your code produces the expected output. Another option would be to use a hash instead of an array as the cache.

A couple more tips:

• Instead of a global, you could define your cache as what is known as a "static" variable in other languages, using state.
• Update: While implementing the cache yourself like that is fine, note that Perl has a module for that, Memoize.
• The &find() style of calling subs is outdated, call your subs without the &.
• The if($num == 0) check isn't necessary if you pre-populate your array with that index.
• You can use Test::More instead of inputting your test cases manually:

use warnings;
use strict;
use feature 'state';

sub find {
    state @cache = (1);
    my $num = shift;
    return 0 if $num < 0;
    if ( defined $cache[$num] ) { return $cache[$num] }
    $cache[$num] = find($num-1) + find($num-2) + find($num-3);
    return $cache[$num];
}

use Test::More tests=>3;
is find(2), 2;
is find(3), 4;
is find(4), 7;
[download]