To save yourself some grief, I would simply look for > and then replace the line, regardless of what else is there (which is what you are doing anyway). You can also use  print $variable and sleep(3); in parts of the code to help see what is present in certain places, particularly if you don't get a match. We don't have the input files, so we can't tell if there are spaces or tabs in the FASTA headers.

use strict;
use warnings;

my @files = glob("*.fa");

for my $file ( @files ) {
   # open the input file
   open my $in, "<", "$file" or die "Cannot open "$file": $!\n";
   # open the output file
   open my $out, ">", "$file.out" or die "Cannot open "$file.out": $!\
+n";
   # reset contig number 
   my $contig_number = 1;

   while ( <$in> ) {
      chomp;
      if ( $_ ~= m/^>/ ) {
         # it's a header
         print $out ">config_$contig_number\n\n";
         $contig_number++;
      }
      else {
         # it's sequence
         print $out "$_\n\n";
      }
      close $in;
      close $out;
   }
}

my $file_count = @files;
print "Successfully processed $file_count files!\n";
exit;
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