Quick Answer

Option 1

import requests
s = requests.session()
s.cookies.set("COOKIE_NAME", "the cookie works", domain="example.com")

Option 2

import requests
s = requests.session()
# Note that domain keyword parameter is the only optional parameter here
cookie_obj = requests.cookies.create_cookie(domain="example.com",name="COOKIE_NAME",value="the cookie works")
s.cookies.set_cookie(cookie_obj)

Detailed Answer

I do not know if this technique was valid when the original question was asked, but ideally you would generate your own cookie object using requests.cookies.create_cookie(name,value,**kwargs) and then add it to the cookie jar via requests.cookies.RequestsCookieJar.set_cookie(cookie,*args,**kwargs). See the source/documentation here.

Adding a custom cookie to requests session

>>> import requests
>>> s = requests.session()
>>> print(s.cookies)
<RequestsCookieJar[]>
>>> required_args = {
        'name':'COOKIE_NAME',
        'value':'the cookie works'
    }
>>> optional_args = {
    'version':0,
    'port':None,
#NOTE: If domain is a blank string or not supplied this creates a
# "super cookie" that is supplied to all domains.
    'domain':'example.com',
    'path':"https://stackoverflow.com/",
    'secure':False,
    'expires':None,
    'discard':True,
    'comment':None,
    'comment_url':None,
    'rest':{'HttpOnly': None},
    'rfc2109':False
}
>>> my_cookie = requests.cookies.create_cookie(**required_args,**optional_args)
# Counter-intuitively, set_cookie _adds_ the cookie to your session object,
#  keeping existing cookies in place
>>> s.cookies.set_cookie(my_cookie)
>>> s.cookies
<RequestsCookieJar[Cookie(version=0, name="COOKIE_NAME", value="the cookie works", port=None, port_specified=False, domain='www.domain.com', domain_specified=True, domain_initial_dot=False, path="https://stackoverflow.com/", path_specified=True, secure=False, expires=None, discard=True, comment=None, comment_url=None, rest={'HttpOnly': None}, rfc2109=False)]>

Bonus: Lets add a super cookie then delete it

>>> my_super_cookie = requests.cookies.create_cookie('super','cookie')
>>> s.cookies.set_cookie(my_super_cookie)
# Note we have both our previous cookie and our new cookie
>>> s.cookies
<RequestsCookieJar[Cookie(version=0, name="super", value="cookie", port=None, port_specified=False, domain='', domain_specified=False, domain_initial_dot=False, path="https://stackoverflow.com/", path_specified=True, secure=False, expires=None, discard=True, comment=None, comment_url=None, rest={'HttpOnly': None}, rfc2109=False), Cookie(version=0, name="COOKIE_NAME", value="the cookie works", port=None, port_specified=False, domain='www.domain.com', domain_specified=True, domain_initial_dot=False, path="https://stackoverflow.com/", path_specified=True, secure=False, expires=None, discard=True, comment=None, comment_url=None, rest={'HttpOnly': None}, rfc2109=False)]>
# Deleting is simple, note that this deletes the cookie based on the name,
# if you have multiple cookies with the same name it will raise
# requests.cookies.CookieConflictError
>>> del s.cookies['super']
>>> s.cookies
<RequestsCookieJar[Cookie(version=0, name="COOKIE_NAME", value="the cookie works", port=None, port_specified=False, domain='www.domain.com', domain_specified=True, domain_initial_dot=False, path="https://stackoverflow.com/", path_specified=True, secure=False, expires=None, discard=True, comment=None, comment_url=None, rest={'HttpOnly': None}, rfc2109=False)]>