I haven't worked with XML::DOM before (I usually use XML::Smart), but here's a working example that extracts the zipcode field from the example data you provided:

use strict;
use warnings;
use XML::DOM;

my $xml = '';
$xml .= $_ for <DATA>;


my $parser    = XML::DOM::Parser->new;
my $doc       = $parser->parse($xml);
my $result_list  = $doc->getElementsByTagName('Result');
my $result_count = $result_list->getLength;

for my $result_n (0..($result_count - 1)) {
    my $result   = $result_list->item($result_n);
    my $zip_nodes = $doc->getElementsByTagName('Zip');
    my $zip = $zip_nodes->item(0)->getChildNodes->item(0)->getData;
    print "zipcode: $zip\n";
}


__DATA__
<?xml version="1.0" encoding="UTF-8"?>
<ResultSet>
  <Result precision="address">
    <Latitude>37.416384</Latitude>
    <Longitude>-122.024853</Longitude>
    <Address>701 FIRST AVE</Address>
    <City>SUNNYVALE</City>
    <State>CA</State>
    <Zip>94089-1019</Zip>
    <Country>US</Country>
  </Result>
</ResultSet>
[download]

It should have some additional error handling for malformed documents, but that's easy enough to add.