That space (paragraph mark) is an intrinsic part of table creation. You cannot get rid of it. However, you can change its size. Turn on display of paragraph marks and other formatting.
Showing non-printing formatting marks in Microsoft Word
Go into the header and select that paragraph mark.
Go to the Home Tab and in the font size type 1 and press Enter.
You may want to get rid of the top border on the table you want to be flush with the header. It won't be flush, but it will be very close.