You have to set the CURLOPT_HEADERDATA and CURLOPT_WRITEDATA options. My testing indicates that their default value is STDOUT:

my $response_data;
my $response_header;
open(my $RD, ">", \$response_data);
open(my $RH, ">", \$response_header);
...
$curl->setopt(CURLOPT_WRITEDATA, $RD);
$curl->setopt(CURLOPT_HEADERDATA, $RH);
...
$curl->perform;
print "Response header: $response_header\n";
print "Response body: $response_body\n";
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Unforunately it doesn't seem you can use undef for these options.

Some more testing indicates that CURLOPT_HEADER option determines whether or not the headers are copied through the CURLTOP_WRITEDATA handle as well as through the CURLOPT_HEADERDATA handle.