There are a few mitigations but my favourite is the "casino chips" approach: pregenerate them server side, and allocate to clients on demand, including en masse if need be ("here kid, have a few million UUIDs to get you started"). Verify with whatever simple signature scheme comes with your application server framework, or at small scale just toss them in a crude LRU store.
Or, remember where the UUID came from, and apply their organisational scope to any trust you place upon it. This might work particularly for multi-tenanted SaaS. However it requires that all usage is tenant-bounded end-through-end throughout your application. This may be in conflict with a) your framework, b) your zenlike contemplation of simplicity in data management, or c) programmers in a hurry forgetting to scope their queries properly.
Ultimately, relying on UUIDs as intrinsically unguessable security tokens is probably not a great idea, but it's one that remains thoroughly embedded in the programming zeitgeist. As ever, nothing useful comes without a compromise. Keep your eyes open to the systemic consequences of design choices, and don't leave traps for your fellow developers.