diff --git a/Lib/tarfile.py b/Lib/tarfile.py
index 721f9d7..999a99b 100755
--- a/Lib/tarfile.py
+++ b/Lib/tarfile.py
@@ -1549,7 +1549,9 @@
 
         if mode in ("r", "r:*"):
             # Find out which *open() is appropriate for opening the file.
-            for comptype in cls.OPEN_METH:
+            def not_compressed(comptype):
+                return cls.OPEN_METH[comptype] == 'taropen'
+            for comptype in sorted(cls.OPEN_METH, key=not_compressed):
                 func = getattr(cls, cls.OPEN_METH[comptype])
                 if fileobj is not None:
                     saved_pos = fileobj.tell()
diff --git a/Lib/test/test_tarfile.py b/Lib/test/test_tarfile.py
index abfb34d..1efb841 100644
--- a/Lib/test/test_tarfile.py
+++ b/Lib/test/test_tarfile.py
@@ -3,6 +3,7 @@
 import io
 from hashlib import md5
 from contextlib import contextmanager
+from random import Random
 
 import unittest
 import unittest.mock
@@ -349,12 +350,17 @@
 
     def test_ignore_zeros(self):
         # Test TarFile's ignore_zeros option.
+        # generate 512 pseudorandom bytes
+        data = Random(0).getrandbits(512*8).to_bytes(512, 'big')
         for char in (b'\0', b'a'):
             # Test if EOFHeaderError ('\0') and InvalidHeaderError ('a')
             # are ignored correctly.
             with self.open(tmpname, "w") as fobj:
                 fobj.write(char * 1024)
-                fobj.write(tarfile.TarInfo("foo").tobuf())
+                tarinfo = tarfile.TarInfo("foo")
+                tarinfo.size = len(data)
+                fobj.write(tarinfo.tobuf())
+                fobj.write(data)
 
             tar = tarfile.open(tmpname, mode="r", ignore_zeros=True)
             try:
diff --git a/Misc/NEWS b/Misc/NEWS
index f7226fd..298d3aa 100644
--- a/Misc/NEWS
+++ b/Misc/NEWS
@@ -113,6 +113,10 @@
 Library
 -------
 
+- Issue #28449: tarfile.open() with mode "r" or "r:" now tries to open a tar
+  file with compression before trying to open it without compression.  Otherwise
+  it had 50% chance failed with ignore_zeros=True.
+
 - Issue #23262: The webbrowser module now supports Firefox 36+ and derived
   browsers.  Based on patch by Oleg Broytman.