You can do it like this:
On your Inbox page, click the Selected button on the top right.
Then, at the bottom of the page a blue bar will appear.
Click Select all on the blue bar, and then in the pulldown menu pick Mark as read.
The only problem is that you have to do it for every page of messages (by clicking "Next"), as far as I can see. So if you have a lot, it could take some time.