Your name space isn't poluted:

use strict;
use warnings;

use mymodule;

print "My Test\n";
Func1();             # ERROR!
Func2("Fluffy", 5);  # ERROR!
[download]

But check this out:

use strict;
use warnings;

use mymodule qw( Func1 );   # Import Func1

#use mymodule qw( Func2 );  # This won't work since
                            # Func2 is not in @EXPORT_OK.

print "My Test\n";
Func1();
Func2("Fluffy", 5);  # ERROR!
[download]

You should never export methods. As you can see, there's no reason to do so. All functions in a package are tied to the object blessed to that package, whether you use Exporter or not.

and how can I prevent it as it seems to be allowing everything into the calling script's namespace?

It's possible to disguise a function using lexicals and/or closures so that noone outside the package can call it, but there's no need to do so. Write proper documentation instead.