diff --git a/Lib/asyncio/taskgroups.py b/Lib/asyncio/taskgroups.py
index 9be4838e3c7ad0..5d5e2a8a85dd48 100644
--- a/Lib/asyncio/taskgroups.py
+++ b/Lib/asyncio/taskgroups.py
@@ -116,10 +116,9 @@ async def __aexit__(self, et, exc, tb):
         if self._base_error is not None:
             raise self._base_error
 
-        if propagate_cancellation_error is not None:
-            # The wrapping task was cancelled; since we're done with
-            # closing all child tasks, just propagate the cancellation
-            # request now.
+        # Propagate CancelledError if there is one, except if there
+        # are other errors -- those have priority.
+        if propagate_cancellation_error and not self._errors:
             raise propagate_cancellation_error
 
         if et is not None and et is not exceptions.CancelledError:
diff --git a/Lib/test/test_asyncio/test_taskgroups.py b/Lib/test/test_asyncio/test_taskgroups.py
index 74bae06af8e729..6a0231f2859a62 100644
--- a/Lib/test/test_asyncio/test_taskgroups.py
+++ b/Lib/test/test_asyncio/test_taskgroups.py
@@ -230,29 +230,29 @@ async def runner():
 
         self.assertEqual(NUM, 15)
 
-    async def test_cancellation_in_body(self):
+    async def test_taskgroup_08(self):
 
         async def foo():
-            await asyncio.sleep(0.1)
-            1 / 0
+            try:
+                await asyncio.sleep(10)
+            finally:
+                1 / 0
 
         async def runner():
             async with taskgroups.TaskGroup() as g:
                 for _ in range(5):
                     g.create_task(foo())
 
-                try:
-                    await asyncio.sleep(10)
-                except asyncio.CancelledError:
-                    raise
+                await asyncio.sleep(10)
 
         r = asyncio.create_task(runner())
         await asyncio.sleep(0.1)
 
         self.assertFalse(r.done())
         r.cancel()
-        with self.assertRaises(asyncio.CancelledError) as cm:
+        with self.assertRaises(ExceptionGroup) as cm:
             await r
+        self.assertEqual(get_error_types(cm.exception), {ZeroDivisionError})
 
     async def test_taskgroup_09(self):
 
@@ -316,8 +316,10 @@ async def runner():
     async def test_taskgroup_11(self):
 
         async def foo():
-            await asyncio.sleep(0.1)
-            1 / 0
+            try:
+                await asyncio.sleep(10)
+            finally:
+                1 / 0
 
         async def runner():
             async with taskgroups.TaskGroup():
@@ -325,24 +327,26 @@ async def runner():
                     for _ in range(5):
                         g2.create_task(foo())
 
-                    try:
-                        await asyncio.sleep(10)
-                    except asyncio.CancelledError:
-                        raise
+                    await asyncio.sleep(10)
 
         r = asyncio.create_task(runner())
         await asyncio.sleep(0.1)
 
         self.assertFalse(r.done())
         r.cancel()
-        with self.assertRaises(asyncio.CancelledError):
+        with self.assertRaises(ExceptionGroup) as cm:
             await r
 
+        self.assertEqual(get_error_types(cm.exception), {ExceptionGroup})
+        self.assertEqual(get_error_types(cm.exception.exceptions[0]), {ZeroDivisionError})
+
     async def test_taskgroup_12(self):
 
         async def foo():
-            await asyncio.sleep(0.1)
-            1 / 0
+            try:
+                await asyncio.sleep(10)
+            finally:
+                1 / 0
 
         async def runner():
             async with taskgroups.TaskGroup() as g1:
@@ -352,19 +356,19 @@ async def runner():
                     for _ in range(5):
                         g2.create_task(foo())
 
-                    try:
-                        await asyncio.sleep(10)
-                    except asyncio.CancelledError:
-                        raise
+                    await asyncio.sleep(10)
 
         r = asyncio.create_task(runner())
         await asyncio.sleep(0.1)
 
         self.assertFalse(r.done())
         r.cancel()
-        with self.assertRaises(asyncio.CancelledError):
+        with self.assertRaises(ExceptionGroup) as cm:
             await r
 
+        self.assertEqual(get_error_types(cm.exception), {ExceptionGroup})
+        self.assertEqual(get_error_types(cm.exception.exceptions[0]), {ZeroDivisionError})
+
     async def test_taskgroup_13(self):
 
         async def crash_after(t):
@@ -424,8 +428,9 @@ async def runner():
 
         self.assertFalse(r.done())
         r.cancel()
-        with self.assertRaises(asyncio.CancelledError):
+        with self.assertRaises(ExceptionGroup) as cm:
             await r
+        self.assertEqual(get_error_types(cm.exception), {ZeroDivisionError})
 
     async def test_taskgroup_16(self):
 
@@ -451,8 +456,9 @@ async def runner():
 
         self.assertFalse(r.done())
         r.cancel()
-        with self.assertRaises(asyncio.CancelledError):
+        with self.assertRaises(ExceptionGroup) as cm:
             await r
+        self.assertEqual(get_error_types(cm.exception), {ZeroDivisionError})
 
     async def test_taskgroup_17(self):
         NUM = 0
diff --git a/Misc/NEWS.d/next/Library/2022-08-08-01-42-11.gh-issue-95704.MOPFfX.rst b/Misc/NEWS.d/next/Library/2022-08-08-01-42-11.gh-issue-95704.MOPFfX.rst
new file mode 100644
index 00000000000000..31f9fc6547d90f
--- /dev/null
+++ b/Misc/NEWS.d/next/Library/2022-08-08-01-42-11.gh-issue-95704.MOPFfX.rst
@@ -0,0 +1,2 @@
+When a task catches :exc:`asyncio.CancelledError` and raises some other error,
+the other error should generally not silently be suppressed.