I would probably have written something like:

use strict;
use warnings;
use Data::Dumper;

my %hash = ( a => 1, b => 0, c => undef, e => 1, f => 0, g => undef );

my @interesting_keys = qw( a b c d );

my %small_hash = map {
     ($hash{$_}) ? ($_ => $hash{$_}) : ()
} @interesting_keys;

print Dumper(\%small_hash);
[download]

but maybe the following is easier to understand...

my %small_hash =
        map { $_ => $hash{$_} }
        grep { $hash{$_} }
        @interesting_keys;
[download]

Both give the same result:

$VAR1 = {
          'a' => 1
        };
[download]

which is what you are wanting, if I understood correctly.

These are only a little different from some of the other suggestions you have received and I don't suggest they are necessarily any better. Which is best for you depends on what you and the others that might read you code know of Perl. You should choose a solution which you find easy to understand and modify or re-use as your needs evolve.