LWP handles digest authentication as long as you get the credentials right. A simple example...
my $ua = LWP::UserAgent->new;
$ua->credentials('jigsaw.w3.org:80', 'test', 'guest', 'guest');
my $response = $ua->get('http://jigsaw.w3.org/HTTP/Digest/');
$response->is_success or die $response->status_line;