Rapid evolution did happen. Look at pygmy animals. They evolved to a smaller scale at such a rapid pace, that they produced several genetic abnormalities that didn't help them survive.
Ask the dinosaurs.
It's debatable that the dinosaurs were killed by an asteroid; massive environmental changes didn't kill the woolly mammoth, afterall. It survived for a good ten thousand years after the global temperature increased. The argument that dinosaurs were so simply killed by an asteroid seems... like a quick answer, considering that not a whole lot else died except dinosaurs.
A few things about evolution and mass extinction. Creatures evolve because of behavior; the alteration in behavior selects individual with certain characteristics from each, and each moves downward until new niches are created. Other times, positive attributes are selected while negative are not; red hair led to blonde hair in humans, it was selected because creatures with blonde hair appeared more attractive. Birds mate because of good song and good looks; a male bird that sing healthily and has bright, colorful plumage shows the female that it can take care of itself (certain foods are required to get that plumage, being able to find it implies they are good eaters and provide well). Not all evolution is necessity based. Also consider that the cohabitation of birds and small, feral animals with humans. Those animals which don't attack humans tend to survive, while those that attack don't. Selecting behavior patterns helps them survive.
So, dinosaurs killed by asteroid? Hardly. Dinosaurs died because, in light of dramatic ecological changes, they could not behave in an advantageous way. Species were too specialized, and could not alter it. The woolly mammoth did, but we killed it.
Now, back to rapid evolution like in the movie. It's possible that evolution can happen at such a fast scale, provided that it's already programmed in there. The creatures that landed on Earth are hard-wired to evolve and adapt as fast as possible. The DNA is already set up to adapt as readily with enough variation to adapt (and some innovations had to be found, like breathing our oxygen). DNA on Earth is not out of the box. It was built from the ground-up, and had to rebuild itself and reconstruct itself using its base coding to adapt.
It's not even certain that other creatures use DNA. DNA is so highly complex, but makes for such an exceptional transfer of information that it works. Other lifeforms may find different ways, allowing them to adapt in much different, much radical ways (i.e., allowing them to adapt in a single generation, rather than through reproduction).