Yes you can extend the network this way. But you will face half the performance on a bridged network that you will get with a simple wifi network, as well as face reliability issues. I've done this with Airport Express before and faced all those issues myself. Since then I build my networks differently, always with a four port wired router at the core and a hardwired or powerline (if the other router is really a long way away) connection to each router.
Ideally you'd be able to hardwire an ethernet connection to both Airport Express from the older D-Link box. As the Airport Express wifi is unlikely to outrun the 100 BaseT wired on the older D-Link box, you'd enjoy great performance and great reliability with two independent wifi networks.
Both Airport Express can be set to use wired bridge which means all devices on both wifi networks would be using the same DHCP settings and would allow them to communicate directly (even over two different wifi networks). Curiously you'd actually get faster throughput this way than with bridged wifi as both Airport Express would be able to communicate at full speed with their respective devices.