I don't think this is necessarily quicker. But it only ever has one filehandle open.

{
  local $/ = "\\/n";

  my $file_no = 1;
  open my $fh, '>', "file$file_no" or die $!;

  while (<>) {
    print $fh, $_;

    unless ($. % 1000) {
      close $fh;
      $file_no++;
      open $fh, '>', "file$file_no" or die $!;
    }
  }
}
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