When you do use A;, that includes a call to A->import but for A1 there is no import method so no variables are exported. One solution is to give A1 an import method - one that just redirects to A's import method.

package A1;

sub import
{
  A->export_to_level(1, @_);
}
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export_to_level is a special routine provided by Exporter that allows you to control where the symbols will be exported to. The 1 here means export 1 level up. So it's saying "Hey A, don't export your symbols into my package (A1) export them into the package I was called from (B in your example)".

A quick plug for one of my modules. Exporter::Easy lets you write

BEGIN {
  use Exporter();
  use vars qw(@EXPORT @ISA);
  @EXPORT = qw($var1);
  @ISA = qw(Exporter);
}
use vars @EXPORT;
$var1 = "1";
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as

use Exporter::Easiest q( EXPORT => $var1 );
$var1 = "1";
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it handles setting up all the other Exporter arrays too if you want it.