Maybe it's an even dirtier hack, but this one can speed up things a lot.
Your script saves time by skipping chomp. My addition will save time by avoiding a hidden join for each array interpolation. I found this trick when I was benchmarking anagram algorithms.
I used a 116_246 words dictionary, and I got a significative speed improvement, as you can see from the relative times.
$ wc words.english
 116246  116246  989075 words.english

$ time perl dict.pl words.english # first hack
1.38    user
0.05    system
0:01.43 elapsed
$ time perl dict2.pl words.english  # second hack
0.86     user
0.02     system
0:00.88 elapsed

[download]
Change
push @{$Word{length$_}},$_ while (<>);
# ...
print OUT @{$Word{$_}};

[download]
into 
$Word{length$_} .= $_ while (<>);
# ...
print OUT $Word{$_};

[download]

BTW: ChOas, congratulations for your sanctification!

update
There is always room for improvement, though. ...
Combining the first array solution with the other tricks, we have a further 10% improvement.

#!/usr/bin/perl -w
use strict;
my @words;
my $file;
$words[length$_] .= $_ while (<>);
for (0..$#words)
{
 next unless $words[$_];
 $file=$_-1 . ".mine";
 open OUT,">$file" or die "Cannot open $file.mine: $!\n";
 print OUT $words[$_];
 close OUT;
};

[download]
update (2)
I am afraid that I must have spoiled demerphq's comparison with my previous update. ;)
He very chivalrously included such update and compared the methods shown so far, offering an improvement, that was unfortunately slower than this array + concatenation hack.
Nice Job. Keep up the good work, bro, and many thanks for your analysis.
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(_|| | |(_|><
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In reply to Re: (FASTER) Example of TMTOWTDI by gmax
in thread Extreme Example of TMTOWTDI by Cody Pendant

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