One way that will get you very close is to run a packet capture program on the server (Wireshark, for instance), start a capture and filter for HTTP traffic, access the web site from a client machine, stop the capture and look for the conversation between the client and the server and calculate the time between when the request got to the server and when the response left the server. This should give you the approximate time it took for the server to process the request.
In addition, if you're using IIS on Windows you could enable logging on the web site and look at the logs for a particular client\server session. This will tell you how long it took IIS to process the request. This should get you closer to your goal.