Although the uniparental (or maternal) inheritance of mitochondrial DNA (mtDNA) is widespread, the reasons for its evolution remain unclear. Two main hypotheses have been proposed: selection against individuals containing different mtDNAs (heteroplasmy) and selection against "selfish" mtDNA mutations. Recently, uniparental inheritance was shown to promote adaptive evolution in mtDNA, potentially providing a third hypothesis for its evolution. Here, we explore this hypothesis theoretically and ask if the accumulation of beneficial mutations provides a sufficient fitness advantage for uniparental inheritance to invade a population in which mtDNA is inherited biparentally. In a deterministic model, uniparental inheritance increases in frequency but cannot replace biparental inheritance if only a single beneficial mtDNA mutation sweeps through the population. When we allow successive selective sweeps of mtDNA, however, uniparental inheritance can replace biparental inheritance. Using a stochastic model, we show that a combination of selection and drift facilitates the fixation of uniparental inheritance (compared to a neutral trait) when there is only a single selective mtDNA sweep. When we consider multiple mtDNA sweeps in a stochastic model, uniparental inheritance becomes even more likely to replace biparental inheritance. Our findings thus suggest that selective sweeps of beneficial mtDNA haplotypes can drive the evolution of uniparental inheritance.