I looked up the cookie specification.
When it sends a request to an origin server, the user agent includes a Cookie request header if it has stored cookies that are applicable to the request.
The only conditions that it describes are domain, path, port, and security (https). If you can't restructure your application to avoid this, you're going to get the extraneous cookies, and there's probably nothing you can do about it (short of replacing all the browsers on the Internet, or just not sending cookies to begin with). I'd consider putting a redirect at / to point it to a subdirectory, but that's probably just as much overhead as you're hoping to save, and has semantic implications.
Have you benchmarked it? How big a deal is it exactly? How big of cookies are you sending? Are there other optimizations you could be doing to improve your user experience instead?
Cookie syntax, for reference:
set-cookie = "Set-Cookie2:" cookies
cookies = 1#cookie
cookie = NAME "=" VALUE *(";" set-cookie-av)
NAME = attr
VALUE = value
set-cookie-av = "Comment" "=" value
| "CommentURL" "=" <"> http_URL <">
| "Discard"
| "Domain" "=" value
| "Max-Age" "=" value
| "Path" "=" value
| "Port" [ "=" <"> portlist <"> ]
| "Secure"
| "Version" "=" 1*DIGIT
portlist = 1#portnum
portnum = 1*DIGIT