use Modern::Perl;

open my $fhA, '<', 'FileA.txt' or die $!;
my %hash = map { /(.+)\t(.+)/; $1 => $2 } grep /\S/, <$fhA>;
close $fhA;

open my $fhB, '<', 'FileB.txt' or die $!;
my @output = map { chomp; $hash{$_} } grep /\S/, <$fhB>;
close $fhB;

open my $fhO, '>', 'Output.txt' or die $!;
say $fhO $_ for @output;
close $fhO;