Mitochondria are not produced de novo in newly divided daughter cells, but are inherited from the mother cell during mitosis. While mitochondrial homeostasis is crucial for living cells, the feedback responses that maintain mitochondrial volume across generations of dividing cells remain elusive. Here, using a microfluidic yeast 'mother machine', we tracked several generations of fission yeast cells and observed that cell size and mitochondrial volume added exponentially. We discovered that while mitochondrial homeostasis piggybacked on the 'sizer' mechanism of cell size homeostasis, mitochondrial function was a critical determinant of the timing of cell division: cells born with lower than average amounts of mitochondria grew slower and thus added more mitochondria before they divided. Thus, mitochondrial biogenesis was tailored to the volume of mitochondria at birth, such that all cells ultimately contained the same mitochondrial volume at cell division. Quantitative modelling and experiments with mitochondrial DNA-deficient rho0 cells additionally revealed that mitochondrial function was essential for driving the exponential growth of cells. Taken together, we demonstrate a central role for mitochondrial activity in dictating cellular growth rates and ensuring mitochondrial volume homeostasis.