Short answer: you don't. But you can mitigate some of the issues.
Even for my own, open-source code, I have a PC sitting here that does nothing but integration testing (unit testing is fine, but is not the panacea many think it is; it's quite possible to say both "all my tests pass!" and "my program doesn't work!"). For web stuff it's got Selenium installed, and exhaustively tests the code to make sure it works sanely. If I have to rush a patch on something, I can at least wait for Selenium to return before pushing it; almost nothing is that time-critical (and if it is, boy did you screw up and it's time to hash out a fast damage-control patch and then go back and do it right). The one caveat is that Selenium and friends (including desktop applications) cannot perform meaningful system testing unless they are on identical hardware to the Real Thing. (For desktop applications, especially consumer-released ones, this kind of system testing is nearly impossible, but given enough money you can build a reasonably broad testing matrix.)
Unit testing is nice and good for fast turnarounds, but I find that you find the really nasty problems in your integration tests.