Your original code seems to work as long as the scalar  $globalVar is exported using its correct name, i.e., using the  $ sigil.

c:\@Work\Perl\monks>perl -wMstrict -le
"MyPackage->import;
 print $::globalVar;
 $::globalVar=1;
 print $::globalVar;
 myfunction();
 BEGIN{
   package MyPackage;
   require Exporter;
   our @ISA = qw(Exporter);
   our @EXPORT = qw($globalVar myfunction);
   our $globalVar = 42;
   sub myfunction { print 'hi from ', __PACKAGE__; }
   }
"
42
1
hi from MyPackage
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