Javascript internally only supports UTF-16.
Because this is an extended 32-bit UTF character (not in the “Basic Multilingual Plane”) you need to insert the “UTF-16 surrogate pair”, which is helpfully provided on the same page that you linked to:
0xD83D 0xDD13
i.e.
$('#myId').text('\ud83d\udd13');
More details can be found in RFC 4627, which is strictly speaking the format for JSON.