It's probable given the right circumstances
The most important thing here is water. Now there's actually quite a lot of water in most deserts, but it's in general hidden beneath the ground. Thus assuming your meteorite/explosion crater had a big enough impact, there's a fair chance that there's now a lake in the center of that crater, fed by underground reserves even.
Lakes? > check (well at least a big one in the center)
Now that we've got water we need to ensure it does not just evaporate and get carried away from our crater-oasis. Assuming a big enough impact we should have walls/sides tall enough to force at least part of the evaporated water to fall in the form of rain again. Thus we get a distribution of the water over a bigger area as needed (alas unless it's been a truly tremendous impact, some of the water will be lost and carried away permanently..)
Rivers? -> probably (there will at least be streams and brooks)
Now the mountains and hills thing is something else. As this is a crater made by an explosion, it will resemble a huge bowl most likely. So tall hills and little mountains will be restricted to the rim of it, alas these walls could be as tall as you manage to get them, so there's your mountains.
Additional thoughts thanks to @Zxyrra: As we've got a weather-cycle going, we can add more lakes & ravines coming from the crater sides towards the central recess; thanks to erosion.
Mountains? Hills? >
well, we have crater walls. That counts for something, right? sure!, just make the whole thing old enough.
So there you go. Having a forest & grasslands inside a huge crater is definitely probable; Though the real question is how you got that crater in the first place?. You know, without blowing the planet up or at least destroying all life on it.