I found two issues with that perl code. You need the first 80 bytes, not 0x80. Second, calculate back to big-endian to get the same result as the miner:
#!/usr/bin/perl
use Digest::SHA qw/ sha256 /;
$raw = pack 'H*', '00000001258124a1e0837367309ed9433af69c741513067793bf1f490000c0c800000000f2a45d9b1294bf78d27fe1d77558fbedf2b1eb37bb5f1808d7b77e33d809b8fb4d1c26d21b04864c01000000000000800000000000000000000000000000000000000000000000000000000000000000000000000000000080020000';
$short = substr($raw, 0, 80); # first 80 byte
$short_le = pack "N*", unpack "V*", $short; # to Little Endian
print "short_le:", unpack("H*", $short_le), "\n";
$hash1 = sha256($short_le);
$hash = sha256(sha256($short_le));
$hash1_be = pack "N*", unpack("V*", $hash1); # BE again
$hash_be = pack "N*", unpack("V*", $hash);
print "hash1: ", unpack("H*", $hash1_be), "\n";
print "hash: ", unpack("H*", $hash_be), "\n";
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