:platively new to real Perl programming, and I'm sure, as always, there are many ways, but one way:

#!/usr/bin/perl -wT

use strict;

#my $filename is somehow parsed before this
$filename =~ /^(\w+)\.\w+$/;
my $replacewith = $1;

open (FILE, "<$filename") || die ("Error: $!");
   my $line = <FILE>;
   chomp $line;
   my $toreplace = substr($line,-10);
close FILE;

$line =~ s/$toreplace/$replacewith/;
[download]

You did not mention whether the new file name will be 10 characters.

Update
Thanks hena for the elegant rewriting of the new file. Forgot that part!