If you want an internal solution, you could try something like writing a short sub or module which redirects STDOUT and STDERR into files, e.g

use FindBin        ();
use File::Basename ();

sub RedirectOutputToFiles {
  my $scriptName = File::Basename::basename($0);
  my $stdoutFile = "$FindBin::Bin/$scriptName.stdout";
  my $stderrFile = "$FindBin::Bin/$scriptName.stderr";

  open (STDOUT, ">", $stdoutFile)
    or die "Error: couldn't redirect STDOUT: '$stdoutFile': $!";

  open (STDERR, ">", $stderrFile)
    or die "Error: couldn't redirect STDERR: '$stderrFile': $!";

} # RedirectOutputToFiles
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or the like, and then just execute the sub somehow