diff --git a/Lib/tarfile.py b/Lib/tarfile.py
index 63c7554af20..3d52a81b9e1 100644
--- a/Lib/tarfile.py
+++ b/Lib/tarfile.py
@@ -1804,11 +1804,13 @@ class TarFile(object):
             fileobj = gzip.GzipFile(name, mode + "b", compresslevel, fileobj)
             t = cls.taropen(name, mode, fileobj, **kwargs)
         except IOError:
-            if not extfileobj:
+            if not extfileobj and fileobj is not None:
                 fileobj.close()
+            if fileobj is None:
+                raise
             raise ReadError("not a gzip file")
         except:
-            if not extfileobj:
+            if not extfileobj and fileobj is not None:
                 fileobj.close()
             raise
         t._extfileobj = extfileobj
diff --git a/Lib/test/test_tarfile.py b/Lib/test/test_tarfile.py
index 8036c5cda13..b45f1689901 100644
--- a/Lib/test/test_tarfile.py
+++ b/Lib/test/test_tarfile.py
@@ -1682,6 +1682,14 @@ class LinkEmulationTest(ReadTest):
 class GzipMiscReadTest(MiscReadTest):
     tarname = gzipname
     mode = "r:gz"
+
+    def test_non_existent_targz_file(self):
+        # Test for issue11513: prevent non-existent gzipped tarfiles raising
+        # multiple exceptions.
+        with self.assertRaisesRegex(IOError, "xxx") as ex:
+            tarfile.open("xxx", self.mode)
+        self.assertEqual(ex.exception.errno, errno.ENOENT)
+
 class GzipUstarReadTest(UstarReadTest):
     tarname = gzipname
     mode = "r:gz"
diff --git a/Misc/NEWS b/Misc/NEWS
index 354d09a070c..328ec61c1ea 100644
--- a/Misc/NEWS
+++ b/Misc/NEWS
@@ -44,6 +44,9 @@ Core and Builtins
 Library
 -------
 
+- Issue #11513: Fix exception handling ``tarfile.TarFile.gzopen()`` when
+  the file cannot be opened.
+
 - Issue #12687: Fix a possible buffering bug when unpickling text mode
   (protocol 0, mostly) pickles.