The built-in glob splits its arguments on whitespace, so one might think of this as a possibility, but especially since you seem to be on Windows, I wouldn't be surprised if the pathnames contain spaces. I would recommend bsd_glob from File::Glob instead, which does not split on whitespace:
use File::Glob 'bsd_glob';
my @files = ( bsd_glob("$Dir1/*"), bsd_glob("$Dir2/*") );
You can also use File::Glob ':bsd_glob'; to override the builtin glob (except on really old versions of Perl). Another possible caveat of glob to be aware of is that it does not list files beginning with a dot by default.
Update: Another possibility might be: my @files = bsd_glob("{$Dir1,$Dir2}/*"); Also a few edits to add a bit of info.
Update 2: An even bigger possible caveat that should be mentioned is that of course all this assumes that $Dir1 and $Dir2 don't contain any characters that have special meaning to glob. If that is the case, I'd recommend other methods like Path::Class::Dir's children method (this will include filenames beginning with a dot, use ->children(no_hidden => 1) to suppress them):
use Path::Class qw/dir/;
my @files = ( dir($Dir1)->children, dir($Dir2)->children );
