Essential genes tend to be highly conserved across eukaryotes, but, in some cases, their critical roles can be bypassed through genetic rewiring. From a systematic analysis of 728 different essential yeast genes, we discovered that 124 (17%) were dispensable essential genes. Through whole-genome sequencing and detailed genetic analysis, we investigated the genetic interactions and genome alterations underlying bypass suppression. Dispensable essential genes often had paralogs, were enriched for genes encoding membrane-associated proteins, and were depleted for members of protein complexes. Functionally related genes frequently drove the bypass suppression interactions. These gene properties were predictive of essential gene dispensability and of specific suppressors among hundreds of genes on aneuploid chromosomes. Our findings identify yeast's core essential gene set and reveal that the properties of dispensable essential genes are conserved from yeast to human cells, correlating with human genes that display cell line-specific essentiality in the Cancer Dependency Map (DepMap) project.