diff --git a/docs/howto/upgrade.rst b/docs/howto/upgrade.rst
index db6bf11f..8cfd7b4b 100644
--- a/docs/howto/upgrade.rst
+++ b/docs/howto/upgrade.rst
@@ -259,9 +259,12 @@ Arguments of :func:`~asyncio.client.connect`
 ``extra_headers`` → ``additional_headers``
 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 
-If you're adding headers to the handshake request sent by
-:func:`~legacy.client.connect` with the ``extra_headers`` argument, you must
-rename it to ``additional_headers``.
+If you're setting the ``User-Agent`` header with the ``extra_headers`` argument,
+you should set it with ``user_agent_header`` instead.
+
+If you're adding other headers to the handshake request sent by
+:func:`~legacy.client.connect` with ``extra_headers``, you must rename it to
+``additional_headers``.
 
 Arguments of :func:`~asyncio.server.serve`
 ..........................................
@@ -310,7 +313,10 @@ replace it with a ``process_request`` function or coroutine.
 ``extra_headers`` → ``process_response``
 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 
-If you're adding headers to the handshake response sent by
+If you're setting the ``Server`` header with ``extra_headers``, you should set
+it with the ``server_header`` argument instead.
+
+If you're adding other headers to the handshake response sent by
 :func:`~legacy.server.serve` with the ``extra_headers`` argument, you must write
 a ``process_response`` callable instead.
 
diff --git a/src/websockets/asyncio/client.py b/src/websockets/asyncio/client.py
index 9582a4bb..1e560fe0 100644
--- a/src/websockets/asyncio/client.py
+++ b/src/websockets/asyncio/client.py
@@ -89,7 +89,7 @@ async def handshake(
             if additional_headers is not None:
                 self.request.headers.update(additional_headers)
             if user_agent_header:
-                self.request.headers["User-Agent"] = user_agent_header
+                self.request.headers.setdefault("User-Agent", user_agent_header)
             self.protocol.send_request(self.request)
 
         await asyncio.wait(
diff --git a/src/websockets/sync/client.py b/src/websockets/sync/client.py
index e2a28764..b7ab8366 100644
--- a/src/websockets/sync/client.py
+++ b/src/websockets/sync/client.py
@@ -86,7 +86,7 @@ def handshake(
             if additional_headers is not None:
                 self.request.headers.update(additional_headers)
             if user_agent_header is not None:
-                self.request.headers["User-Agent"] = user_agent_header
+                self.request.headers.setdefault("User-Agent", user_agent_header)
             self.protocol.send_request(self.request)
 
         if not self.response_rcvd.wait(timeout):
diff --git a/tests/asyncio/test_client.py b/tests/asyncio/test_client.py
index bdd519fb..9c7ee46a 100644
--- a/tests/asyncio/test_client.py
+++ b/tests/asyncio/test_client.py
@@ -121,6 +121,14 @@ async def test_remove_user_agent(self):
             async with connect(get_uri(server), user_agent_header=None) as client:
                 self.assertNotIn("User-Agent", client.request.headers)
 
+    async def test_legacy_user_agent(self):
+        """Client can override User-Agent header with additional_headers."""
+        async with serve(*args) as server:
+            async with connect(
+                get_uri(server), additional_headers={"User-Agent": "Smith"}
+            ) as client:
+                self.assertEqual(client.request.headers["User-Agent"], "Smith")
+
     async def test_keepalive_is_enabled(self):
         """Client enables keepalive and measures latency by default."""
         async with serve(*args) as server:
diff --git a/tests/sync/test_client.py b/tests/sync/test_client.py
index dbecadca..4844d3b5 100644
--- a/tests/sync/test_client.py
+++ b/tests/sync/test_client.py
@@ -82,6 +82,14 @@ def test_remove_user_agent(self):
             with connect(get_uri(server), user_agent_header=None) as client:
                 self.assertNotIn("User-Agent", client.request.headers)
 
+    def test_legacy_user_agent(self):
+        """Client can override User-Agent header with additional_headers."""
+        with run_server() as server:
+            with connect(
+                get_uri(server), additional_headers={"User-Agent": "Smith"}
+            ) as client:
+                self.assertEqual(client.request.headers["User-Agent"], "Smith")
+
     def test_keepalive_is_enabled(self):
         """Client enables keepalive and measures latency by default."""
         with run_server() as server: