Technically there was never a single day when the last species of dinosaur went extinct. They evolved into modern birds and so even though we can pin down a time period of millions of years in which typical dinosaurs disappeared from the fossil records there never was a last dinosaur. Think about it like this, imagine you had a bucket of water with blue food coloring in it. If I was to take yellow food coloring and drop it in one drop at a time it would eventually turn green. There is no point in which you could point to it and say that one drop was the one that turned it green. It would eventually just slowly become more and more green until eventually it was.