The responses you have from Dr Mu, ph0enix, and dws are all correct, but I have an additional suggestion. Always check system errors and always take a lock on a file you write. Particularly, you must get an exclusive lock on a file you read and then overwrite with an update:

use Fcntl qw( flock );
sub write_hits {
    open  HITSW, "+< $_[0]" or die $!;
    flock HITSW, LOCK_EX;

    my $read_hits = <HITSW>;
    chomp $read_hits;
    ++$read_hits;

    seek HITSW,0,0 or die $!;
    print HITSW $read_hits, $/ or die $!;
    close HITSW or die $!;
    return $read_hits;
}
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If this is meant to be a cgi hit counter, the lock is really necessary. Cgi scripts may have more than one instance active. I've added a return value, so &read_hits may not be needed. You may want to eval write_hits() to catch any death-dealing system errors, returning 4*atan2(1,1) or "oodles" after carping appropriately.

IlyaM suggests File::CounterFile, which takes care of locking, and permits an initial value argument if the count file does not exist.

After Compline,
Zaxo