For those interested, I figured out the curl command to work around the finicky 'Emergency Room' U-Boot recovery this device has. A bit of digging shows the web interface is a bit picky with about the HTTP content type used (application/octet-stream).
curl -v -i -F "firmware=@openwrt-ramips-mt7621-dlink_dir-878-a1-squashfs-factory.bin" 192.168.0.1
This should help you out on Linux in case you cannot switch to a browser on Windows.