Exim expects to receive a full RFC 5322 / 2822 / 822 email message as input. So you need to give the headers in the form of... headers.
$ (echo "To: foo@bar"; echo "Reply-To: my@email.com" echo "Subject: Hello!"; echo ""; echo "My Mail Content") | sendmail foo@bar
$ sendmail foo@bar <<'EOF' From: "The Great Quux" <baz@quux> To: "Fred Foobar" <foo@bar> Reply-To: dev@null Subject: Hello! Content-Type: text/plain; charset=utf-8 My Mail Content EOF
(Not sure if the names in From/To have to be quoted. RFC 5322 seems to say that they should)