Cancer cells generally lack the "safety mechanisms" or "check points" that prevent normal cells from replicating, and some of these mechanisms are also involved in ensuring that DNA replication proceeds without serious errors. Inactivating these safety mechanisms lets cancer to cells divide rapidly, but at the same time also causes their genome replication to be "sloppy", introducing various errors.
The P53 protein is probably the most well-known example of such a "safety mechanism". In normal cells, P53 arrests the cell cycle if breakage occurs in the DNA strands; but cancer cells often inactivate P53, and consequently they tend to proceed with DNA replication even if DNA is damaged. A consequence is that DNA is often in a state of disrepair in cancer cells. (A review of P53 here.)
This lack of proper "quality control" can lead to loss of large pieces of the genome, even entire chromosome arms. While it may seem strange that the cells can survive this, keep in mind that the genome encodes information to support the complex working of an entire organism. Loss of a chromosome arm would indeed be lethal to the organism as a whole --- perhaps it would disrupt brain development, or the immune system. But for single cell to survive and develop, most genes are actually not essential, likely because there are so many redundancies. This is frequenly seen in genome-wide RNA interference screens, for example this paper. Remarkably, cells can even lose the entire mitochondrial DNA (which means they cannot even respire anymore!) and still survive.
It is not clear that "non-cancerous cells would perish" if they would similarly lose DNA. They would likely arrest and turn senescent because they detect DNA damage, precisely because of these safety mechanisms --- but that is different. The loss of genes in itself may not be lethal to a normal cell, it's rather that normal cells "choose" not to continue when this happens. But this distinction is probably difficult to test experimentally.