Actually, in the last few years there was the reverse of that. The government may have a majority in the lower house or a coalition to deal with, and the upper house will have to rely on some pretty strong negotiations with a range of parties to get its legislation through.
There are ministers in both houses. You can be a senator and still be a minister. I can't remember the last time a government had a majority in both the House and the Senate, but it would have been quite a few years ago.