Recent versions indeed support it, but early versions might not. For the March 2008 802.11n version Macworld writes (emphasis mine):
The Express is also the first AirPort Base Station to provide a new feature called ProxySTA. In this mode, the Express acts as a wireless-to-Ethernet bridge, extending your wireless network to wired clients.
Unfortunately, this feature isn’t documented and doesn’t appear by name anywhere in AirPort Utility. To activate ProxySTA, you must set up the Express to join a wireless network and then enable the Allow Ethernet Clients setting; both settings are located in the Wireless tab of AirPort Utility. The feature should work with any wireless network: a, b, g, or n; and 5GHz or 2.4GHz. (Note that the Express, which ships with firmware 7.3, must also be updated to firmware 7.3.1 for this feature to appear.)
Earlier versions seem to support this kind of bridging as well, but your mileage may vary when not using an Apple base station (so, I don't expect problems for your situation). According to Apple's AirPort Express frequently asked questions:
Question: Can AirPort Express wirelessly extend ("repeat" or "rebroadcast") the network of a third-party access point?
Answer: No. But it can wirelessly extend the range of a WDS network that is being hosted by an AirPort Extreme Base Station or another AirPort Express.
Question: Can AirPort Express act as a bridge?
Answer: AirPort Express can act as a bridge in three different ways [..] If it is set up as a WDS remote or relay station, it can bridge the wireless network to wired clients.
Given my problems with a modern but non-Apple base station and an older AirPort Express, I assume a wireless-to-wired bridge needs Apple's WDS capabilities (and hence when reading question 2, I assume one should take question 1 into account as well).
(See also Apple's early 2009 Apple AirPort Networks manual for more information.)