diff --git a/Lib/subprocess.py b/Lib/subprocess.py
index aed7292541e..85b9ea07854 100644
--- a/Lib/subprocess.py
+++ b/Lib/subprocess.py
@@ -489,11 +489,20 @@ def run(*popenargs,
     with Popen(*popenargs, **kwargs) as process:
         try:
             stdout, stderr = process.communicate(input, timeout=timeout)
-        except TimeoutExpired:
+        except TimeoutExpired as exc:
             process.kill()
-            stdout, stderr = process.communicate()
-            raise TimeoutExpired(process.args, timeout, output=stdout,
-                                 stderr=stderr)
+            if _mswindows:
+                # Windows accumulates the output in a single blocking
+                # read() call run on child threads, with the timeout
+                # being done in a join() on those threads.  communicate()
+                # _after_ kill() is required to collect that and add it
+                # to the exception.
+                exc.stdout, exc.stderr = process.communicate()
+            else:
+                # POSIX _communicate already populated the output so
+                # far into the TimeoutExpired exception.
+                process.wait()
+            raise
         except:  # Including KeyboardInterrupt, communicate handled that.
             process.kill()
             # We don't call process.wait() as .__exit__ does that for us.
@@ -1050,12 +1059,16 @@ class Popen(object):
             return endtime - _time()
 
 
-    def _check_timeout(self, endtime, orig_timeout):
+    def _check_timeout(self, endtime, orig_timeout, stdout_seq, stderr_seq,
+                       skip_check_and_raise=False):
         """Convenience for checking if a timeout has expired."""
         if endtime is None:
             return
-        if _time() > endtime:
-            raise TimeoutExpired(self.args, orig_timeout)
+        if skip_check_and_raise or _time() > endtime:
+            raise TimeoutExpired(
+                    self.args, orig_timeout,
+                    output=b''.join(stdout_seq) if stdout_seq else None,
+                    stderr=b''.join(stderr_seq) if stderr_seq else None)
 
 
     def wait(self, timeout=None):
@@ -1843,10 +1856,15 @@ class Popen(object):
                 while selector.get_map():
                     timeout = self._remaining_time(endtime)
                     if timeout is not None and timeout < 0:
-                        raise TimeoutExpired(self.args, orig_timeout)
+                        self._check_timeout(endtime, orig_timeout,
+                                            stdout, stderr,
+                                            skip_check_and_raise=True)
+                        raise RuntimeError(  # Impossible :)
+                            '_check_timeout(..., skip_check_and_raise=True) '
+                            'failed to raise TimeoutExpired.')
 
                     ready = selector.select(timeout)
-                    self._check_timeout(endtime, orig_timeout)
+                    self._check_timeout(endtime, orig_timeout, stdout, stderr)
 
                     # XXX Rewrite these to use non-blocking I/O on the file
                     # objects; they are no longer using C stdio!
diff --git a/Lib/test/test_subprocess.py b/Lib/test/test_subprocess.py
index e58d0925df3..69ac584667d 100644
--- a/Lib/test/test_subprocess.py
+++ b/Lib/test/test_subprocess.py
@@ -10,6 +10,7 @@ import os
 import errno
 import tempfile
 import time
+import traceback
 import selectors
 import sysconfig
 import select
@@ -1557,6 +1558,26 @@ class RunFuncTestCase(BaseTestCase):
         self.assertIn('stderr', c.exception.args[0])
         self.assertIn('capture_output', c.exception.args[0])
 
+    # This test _might_ wind up a bit fragile on loaded build+test machines
+    # as it depends on the timing with wide enough margins for normal situations
+    # but does assert that it happened "soon enough" to believe the right thing
+    # happened.
+    @unittest.skipIf(mswindows, "requires posix like 'sleep' shell command")
+    def test_run_with_shell_timeout_and_capture_output(self):
+        """Output capturing after a timeout mustn't hang forever on open filehandles."""
+        before_secs = time.monotonic()
+        try:
+            subprocess.run('sleep 3', shell=True, timeout=0.1,
+                           capture_output=True)  # New session unspecified.
+        except subprocess.TimeoutExpired as exc:
+            after_secs = time.monotonic()
+            stacks = traceback.format_exc()  # assertRaises doesn't give this.
+        else:
+            self.fail("TimeoutExpired not raised.")
+        self.assertLess(after_secs - before_secs, 1.5,
+                        msg="TimeoutExpired was delayed! Bad traceback:\n```\n"
+                        f"{stacks}```")
+
 
 @unittest.skipIf(mswindows, "POSIX specific tests")
 class POSIXProcessTestCase(BaseTestCase):
diff --git a/Misc/NEWS.d/next/Library/2019-07-04-13-00-20.bpo-37424.0i1MR-.rst b/Misc/NEWS.d/next/Library/2019-07-04-13-00-20.bpo-37424.0i1MR-.rst
new file mode 100644
index 00000000000..b98a17e241e
--- /dev/null
+++ b/Misc/NEWS.d/next/Library/2019-07-04-13-00-20.bpo-37424.0i1MR-.rst
@@ -0,0 +1,5 @@
+Fixes a possible hang when using a timeout on `subprocess.run()` while
+capturing output.  If the child process spawned its own children or
+otherwise connected its stdout or stderr handles with another process, we
+could hang after the timeout was reached and our child was killed when
+attempting to read final output from the pipes.