#!/usr/bin/perl
use warnings;
use strict;

use LWP::UserAgent;
use Cpanel::JSON::XS qw{ encode_json };

my $ua = 'LWP::UserAgent'->new(timeout => 1);
my $response = $ua->post('https://httpbin.org/post',
                         'content-type' => 'application/json',
                         Content => encode_json({key => 'value'}));

print $response->decoded_content;
[download]

I can unequivocally say that what choroba has there works properly. Here's an actual working example I just tested (with some name/data modifications):

use warnings;
use strict;

use LWP::UserAgent;
use JSON;

my $ua = LWP::UserAgent->new;

my $body_content = {
    token => 'NTE2M2Q3YmFmMGEzNjEzMGQyZmFmYmViNGEasdfmNzkzMTE3NzI3MWMz
+MDhkZjliYWE3NTI1OTg1MDQwNjNlZg'
};

my $resp = $ua->post(
    'https://my.server.com',
    'Content-Type'    => 'application/json',
    'Content'         => encode_json $body_content
);

print $resp->decoded_content;
[download]