Kibana won't connect to Elasticsearch on Amazon's Elasticsearch Service

After switching from hosting my own Elastiscsearch cluster to Amazon's Elasticsearch Service, my Kibana dashboards (versions 4.0.2 and 4.1.2) won't load and I'm receiving the following error in kibana.log:

{
  "name": "Kibana",
  "hostname": "logs.example.co",
  "pid": 8037,
  "level": 60,
  "err": {
    "message": "Not Found",
    "name": "Error",
    "stack": "Error: Not Found\n    at respond (\/srv\/kibana\/kibana-4.1.2-linux-x64\/src\/node_modules\/elasticsearch\/src\/lib\/transport.js:235:15)\n    at checkRespForFailure (\/srv\/kibana\/kibana-4.1.2-linux-x64\/src\/node_modules\/elasticsearch\/src\/lib\/transport.js:203:7)\n    at HttpConnector.<anonymous> (\/srv\/kibana\/kibana-4.1.2-linux-x64\/src\/node_modules\/elasticsearch\/src\/lib\/connectors\/http.js:156:7)\n    at IncomingMessage.bound (\/srv\/kibana\/kibana-4.1.2-linux-x64\/src\/node_modules\/elasticsearch\/node_modules\/lodash-node\/modern\/internals\/baseBind.js:56:17)\n    at IncomingMessage.emit (events.js:117:20)\n    at _stream_readable.js:944:16\n    at process._tickCallback (node.js:442:13)"
  },
  "msg": "",
  "time": "2015-10-14T20:48:40.169Z",
  "v": 0
}

unfortunately, this error is not very helpful. I assume it's a wrapped HTTP 404, but for what?

How can I connect a Kibana install to Amazon's Elasticsearch Service?