Instead of relying on the shell background metacharacter like ColtsFoot does, you can use the perl fork function.

use CGI ':standard';

my $pid = fork;
if ($pid == 0) {                    # we are the child
  close;  # close all filehandles so server won't try to stay open
  exec 'backupprogram' or exit  ;  # transfer execution
}
elsif ($pid) {                      # we are the original process
  print header, start_html, 
    'Backup initiated. You can close this window at any time', 
    end_html;
}
else { die "Fork failed: $!\n"; }   # something went wrong
[download]

You could write that as:

fork==0 and exec 'backupprogram';
print header,start_html,'etc...'
[download]

But that wouldn't be as reliable.

Update: Remembered close behaviour wrongly, that line should read
close STDOUT; close STDERR; close STDIN;