That's odd -- it works for me. In a path where "foo.bar" does not exist, running this gives me "I got the error", and nothing else:

perl -e 'open(C,"ls foo.bar 2>&1 |") or die $!; 
 while(<C>){ print "I got the error\n" if (/No such/) }'
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Maybe you need to be sure about the particular shell that is being invoked when you do the pipeline open? (Mine is /bin/bash; I think C-shells are brain-damaged when it comes to redirecting stderr.)