diff --git a/Lib/tarfile.py b/Lib/tarfile.py
index 63c7554..3d52a81 100644
--- a/Lib/tarfile.py
+++ b/Lib/tarfile.py
@@ -1804,11 +1804,13 @@
             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 8036c5c..b45f168 100644
--- a/Lib/test/test_tarfile.py
+++ b/Lib/test/test_tarfile.py
@@ -1682,6 +1682,14 @@
 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"