diff --git a/changelog.d/13957.feature b/changelog.d/13957.feature
new file mode 100644
index 000000000000..4080147357ee
--- /dev/null
+++ b/changelog.d/13957.feature
@@ -0,0 +1 @@
+Ask mail servers receiving emails from Synapse to not send automatic reply (e.g. out-of-office responses).
diff --git a/synapse/handlers/send_email.py b/synapse/handlers/send_email.py
index e2844799e886..804cc6e81e00 100644
--- a/synapse/handlers/send_email.py
+++ b/synapse/handlers/send_email.py
@@ -187,6 +187,19 @@ async def send_email(
         multipart_msg["To"] = email_address
         multipart_msg["Date"] = email.utils.formatdate()
         multipart_msg["Message-ID"] = email.utils.make_msgid()
+        # Discourage automatic responses to Synapse's emails.
+        # Per RFC 3834, automatic responses should not be sent if the "Auto-Submitted"
+        # header is present with any value other than "no". See
+        #     https://www.rfc-editor.org/rfc/rfc3834.html#section-5.1
+        multipart_msg["Auto-Submitted"] = "auto-generated"
+        # Also include a Microsoft-Exchange specific header:
+        #    https://learn.microsoft.com/en-us/openspecs/exchange_server_protocols/ms-oxcmail/ced68690-498a-4567-9d14-5c01f974d8b1
+        # which suggests it can take the value "All" to "suppress all auto-replies",
+        # or a comma separated list of auto-reply classes to suppress.
+        # The following stack overflow question has a little more context:
+        #    https://stackoverflow.com/a/25324691/5252017
+        #    https://stackoverflow.com/a/61646381/5252017
+        multipart_msg["X-Auto-Response-Suppress"] = "All"
         multipart_msg.attach(text_part)
         multipart_msg.attach(html_part)