Replacing small section of ceiling with dry rot

In an atrium overhang area I have dry rot in two adjacent ceiling boards extending no more than 10 inches. These boards span a beam that runs across the atrium. I believe that water ran along the top of the beam and under the board, causing the rot. I can scrape out the rotted wood but figuring out how to patch it is tricky, since the patching material would have to be applied "upside down". If the rot was on the top of an exposed beam, it would be simple.

Has anyone out there dealt with this issue before?

Thanks,
Barry Brisco