diff --git a/requests/sessions.py b/requests/sessions.py
index dd525e2ac9..27d0e9717d 100644
--- a/requests/sessions.py
+++ b/requests/sessions.py
@@ -115,6 +115,22 @@ def get_redirect_target(self, resp):
             return to_native_string(location, 'utf8')
         return None
 
+    def should_strip_auth(self, old_url, new_url):
+        """Decide whether Authorization header should be removed when redirecting"""
+        old_parsed = urlparse(old_url)
+        new_parsed = urlparse(new_url)
+        if old_parsed.hostname != new_parsed.hostname:
+            return True
+        # Special case: allow http -> https redirect when using the standard
+        # ports. This isn't specified by RFC 7235, but is kept to avoid
+        # breaking backwards compatibility with older versions of requests
+        # that allowed any redirects on the same host.
+        if (old_parsed.scheme == 'http' and old_parsed.port in (80, None)
+                and new_parsed.scheme == 'https' and new_parsed.port in (443, None)):
+            return False
+        # Standard case: root URI must match
+        return old_parsed.port != new_parsed.port or old_parsed.scheme != new_parsed.scheme
+
     def resolve_redirects(self, resp, req, stream=False, timeout=None,
                           verify=True, cert=None, proxies=None, yield_requests=False, **adapter_kwargs):
         """Receives a Response. Returns a generator of Responses or Requests."""
@@ -236,14 +252,10 @@ def rebuild_auth(self, prepared_request, response):
         headers = prepared_request.headers
         url = prepared_request.url
 
-        if 'Authorization' in headers:
+        if 'Authorization' in headers and self.should_strip_auth(response.request.url, url):
             # If we get redirected to a new host, we should strip out any
             # authentication headers.
-            original_parsed = urlparse(response.request.url)
-            redirect_parsed = urlparse(url)
-
-            if (original_parsed.hostname != redirect_parsed.hostname):
-                del headers['Authorization']
+            del headers['Authorization']
 
         # .netrc might have more auth for us on our new host.
         new_auth = get_netrc_auth(url) if self.trust_env else None
diff --git a/tests/test_requests.py b/tests/test_requests.py
index fd04ad2705..660437988a 100644
--- a/tests/test_requests.py
+++ b/tests/test_requests.py
@@ -1573,15 +1573,15 @@ def test_nonhttp_schemes_dont_check_URLs(self):
             preq = req.prepare()
             assert test_url == preq.url
 
-    @pytest.mark.xfail(raises=ConnectionError)
-    def test_auth_is_stripped_on_redirect_off_host(self, httpbin):
+    def test_auth_is_stripped_on_http_downgrade(self, httpbin, httpbin_secure, httpbin_ca_bundle):
         r = requests.get(
-            httpbin('redirect-to'),
-            params={'url': 'http://www.google.co.uk'},
+            httpbin_secure('redirect-to'),
+            params={'url': httpbin('get')},
             auth=('user', 'pass'),
+            verify=httpbin_ca_bundle
         )
         assert r.history[0].request.headers['Authorization']
-        assert not r.request.headers.get('Authorization', '')
+        assert 'Authorization' not in r.request.headers
 
     def test_auth_is_retained_for_redirect_on_host(self, httpbin):
         r = requests.get(httpbin('redirect/1'), auth=('user', 'pass'))
@@ -1590,6 +1590,27 @@ def test_auth_is_retained_for_redirect_on_host(self, httpbin):
 
         assert h1 == h2
 
+    def test_should_strip_auth_host_change(self):
+        s = requests.Session()
+        assert s.should_strip_auth('http://example.com/foo', 'http://another.example.com/')
+
+    def test_should_strip_auth_http_downgrade(self):
+        s = requests.Session()
+        assert s.should_strip_auth('https://example.com/foo', 'http://example.com/bar')
+
+    def test_should_strip_auth_https_upgrade(self):
+        s = requests.Session()
+        assert not s.should_strip_auth('http://example.com/foo', 'https://example.com/bar')
+        assert not s.should_strip_auth('http://example.com:80/foo', 'https://example.com/bar')
+        assert not s.should_strip_auth('http://example.com/foo', 'https://example.com:443/bar')
+        # Non-standard ports should trigger stripping
+        assert s.should_strip_auth('http://example.com:8080/foo', 'https://example.com/bar')
+        assert s.should_strip_auth('http://example.com/foo', 'https://example.com:8443/bar')
+
+    def test_should_strip_auth_port_change(self):
+        s = requests.Session()
+        assert s.should_strip_auth('http://example.com:1234/foo', 'https://example.com:4321/bar')
+
     def test_manual_redirect_with_partial_body_read(self, httpbin):
         s = requests.Session()
         r1 = s.get(httpbin('redirect/2'), allow_redirects=False, stream=True)