Backticks interpolate, so you need to escape the dollars and backslashes:

#!/usr/bin/perl -w
use strict;

my $num_subs = `perl -w -0777 -pi -e 'my \$c=s/foo/bar/g;print STDOUT 
+"\$c\\n"' ./foo_test`;

printf("num_subs = $num_subs\n");
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