At least your intuition is accurate, wrt the inefficiency of the code. Not that it's all that bad, and for a script like this, who cares if it runs in 3 seconds or 6?

Here's some suggestions:

• Look at all the places you use sort. Most, if not all, are unnecessary.
• Your reading and parsing of the file could be much more concise, with a payoff in efficiency. To wit:

  while (<FILE>) { # line at a time
    my( $game,$month,$day,$year,$one,$two,$three,$four,$five,$bonus) = s
  +plit;
    for ( $one,$two,$three,$four,$five ) {
      $hash{$_}++;
    }
    $bonus{$bonus}++; # bonus
  }
  [download]

• Your inner "pushing" loops could be replaced with:

    push @draw, ($key) x $hash{$key};
  [download]

  Let perl do the grunt work.
  I might even combine the two "pushing" loops into something like the following-

  foreach my $key ( keys %hash )
  {
      push @draw, ($key) x $hash{$key};
      push @mega, ($key) x $hash{$key}
          if exists $bonus{$key};
  }
  [download]

  since, I'm assuming, a number can only appear in the 'bonus' column if it also appears at least once in one of the other columns.
• Your duplicate-detecting loop has a bug. You want to pick a new number if the current number is the same as any in your @picked array. I assume. Right?
  If so, then you could use grep to do set membership tests on the array.
  Even better would be to use a hash. You're doing many repeated set membership tests on it, and a hash is much better at this than an array. If you care about the order in which numbers are picked (and I assume you do), you could use Tie::IxHash to get an order-preserving hash.
  Or in a case like this you could just use an auxiliary "seen" hash.
  Here's a solution using grep:

      my $number;
      do {
          $number = draw_regular();
      } while ( grep { $_ == $number } @picked );
      push @picked, $number;
  [download]

  Here's a solution using an aux hash:

      my $number;
      do {
          $number = draw_regular();
      } while ( exists $seen{$number} );
      $seen{$number}++;
      push @picked, $number;
  [download]