Eukaryotes carry numerous asexual cytoplasmic genomes (mitochondria and plastids). Lacking recombination, asexual genomes should theoretically suffer from impaired adaptive evolution. Yet, empirical evidence indicates that cytoplasmic genomes experience higher levels of adaptive evolution than predicted by theory. In this study, we use a computational model to show that the unique biology of cytoplasmic genomes---specifically their organization into host cells and their uniparental (maternal) inheritance---enable them to undergo effective adaptive evolution. Uniparental inheritance of cytoplasmic genomes decreases competition between different beneficial substitutions (clonal interference), promoting the accumulation of beneficial substitutions. Uniparental inheritance also facilitates selection against deleterious cytoplasmic substitutions, slowing Muller's ratchet. In addition, uniparental inheritance generally reduces genetic hitchhiking of deleterious substitutions during selective sweeps. Overall, uniparental inheritance promotes adaptive evolution by increasing the level of beneficial substitutions relative to deleterious substitutions. When we assume that cytoplasmic genome inheritance is biparental, decreasing the number of genomes transmitted during gametogenesis (bottleneck) aids adaptive evolution. Nevertheless, adaptive evolution is always more efficient when inheritance is uniparental. Our findings explain empirical observations that cytoplasmic genomes---despite their asexual mode of reproduction---can readily undergo adaptive evolution.