The mitochondrial membrane undergoes extreme remodeling during fission. While a few membrane-squeezing proteins are recognized as the key drivers of fission, there is a growing body of evidence that strongly suggests that conical lipids play a critical role in regulating mitochondrial morphology and fission. However, the mechanisms by which proteins and lipids cooperate to execute fission have not been quantitatively investigated. Here, we computationally model the squeezing of the largely tubular mitochondrion and show that proteins and conical lipids can act synergistically to trigger buckling instability and achieve extreme constriction. More remarkably, the study reveals that the conical lipids can act with different fission proteins to induce hierarchical instabilities and create increasingly narrow and stable constrictions. We reason that this geometric plasticity imparts significant robustness to the fission reaction by arresting the elastic tendency of the membrane to rebound during protein polymerization and depolymerization cycles. Our in vitro study validates protein-lipid cooperativity in constricting membrane tubules. Overall, our work presents a general mechanism for achieving drastic topological remodeling in cellular membranes.