There is one useful package in Perl standard library - CGI::Carp.
It will help you to redirect ALL errors and warnings to your own log file.

BEGIN{
  use CGI::Carp qw(carpout);
  open LOG, $logfile or die "Couldn't open log file: $!";
  carpout(LOG);
}
[download]

see also 'man CGI::Carp';