Gets even better when we acknowledge the fact that our medical definition of "dead" is very convoluted. Dying isn't something that happens all at once, so pinpointing the exact moment someone dies is a bit tricky.

The only thing the heart does to keep the body alive is circulate blood. If you hook soneone up to a machine that keeps their blood pumping, then they don't really need their heart, at least for a time. With modern technology, infection and overall practicality of moving around with such a machine make doing so ineffective in the long run, but it works plenty long enough for surgery.