I've encountered some problems too, as I mentioned. It appears to be a flaky or poorly documented interface. I don't have the time to hunt down the problem, but you could file a bug report and encode manually for now. 
open my $fh, ">", "output.txt" or die $!;
print $fh encode('latin-1', $euro_in_unicode), "\n";
close $fh;

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Update: Taking a hint from the source, the following seems to work. No idea what STOP_AT_PARTIAL means. 

#!/usr/bin/perl
use strict;
use warnings;
use PerlIO::encoding;
use Encode;

my $euro_in_unicode = chr(0x20AC);

open my $fh, ">", "output.txt" or die $!;
{
   local $PerlIO::encoding::fallback =
      Encode::FB_DEFAULT|Encode::STOP_AT_PARTIAL;
   binmode $fh, ":encoding(latin1)";
}
print $fh "$euro_in_unicode\n";
close $fh;

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In reply to Re^3: binmode, encoding layer, ignoring encoding problems by ikegami
in thread binmode, encoding layer, ignoring encoding problems by McA

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