Try this:
import urllib.request
import urllib.parse
import base64
def encode_credentials(username, password):
    byte_creds = '{}:{}'.format(username, password).encode('utf-8')
    return base64.b64encode(byte_creds).decode('utf-8')
def tweet(username, password, message):
    encoded_msg = urllib.parse.urlencode({'status': message})
    credentials = encode_credentials(username, password)
    request = urllib.request.Request(
         'http://api.twitter.com/1/statuses/update.json')
    request.add_header('Authorization', 'Basic ' + credentials)
    urllib.request.urlopen(request, encoded_msg)
Then call tweet('username', 'password', 'Hello twitter from Python3!').
The urlencode function prepares the message for the HTTP POST request.
The Request object uses HTTP authentication as explained here: http://en.wikipedia.org/wiki/Basic_access_authentication
The urlopen methods sends the request to the twitter. When you pass it some data, it uses POST, otherwise GET.
This uses only parts of the Python3 standard library. However, if you want to work with HTTP, you should reconsider using third-party libraries. This Dive Into Python 3 chapter explains how and why.