Because mitochondrial and plastid genomes exist at high cellular copy numbers, new mutations result in the co-occurrence of multiple alleles within a cell or individual, a phenomenon known as heteroplasmy. The extent to which heteroplasmies are transmitted across generations or eliminated through a sorting process involving successive genetic bottlenecks is not well understood in plants, in part because their low mutation rates make these variants so infrequent. We previously found that disruption of MutS Homolog 1 (MSH1), a nuclear gene involved in plant organellar DNA repair, results in numerous de novo point mutations, providing an unprecedented opportunity to explore the inheritance of these variants. Here, we used droplet digital PCR to quantitatively track the inheritance of single nucleotide variants (SNVs) in mitochondrial and plastid genomes of both wild-type Arabidopsis and msh1 mutants. We found that heteroplasmic sorting was rapid for both organelles, greatly exceeding the rates at which mitochondrial heteroplasmies are eliminated in animals. In msh1 mutants, plastid SNVs usually reached fixation or were lost within a single generation, indicating an effective transmission bottleneck size (N) of ~1. In contrast, mitochondrial heteroplasmies often persisted across generations but still exhibited a relatively narrow transmission bottleneck (N ≈ 4). Restoring MSH1 function in mitochondria increased the rate of heteroplasmic sorting (N ≈ 1.3), potentially due to its hypothesized role in promoting gene conversion as a mechanism of DNA repair, which is expected to homogenize genome copies within a cell. We also found that heteroplasmic sorting favored GC base pairs. Therefore, recombinational repair and gene conversion in plant organellar genomes may accelerate the elimination of heteroplasmies and bias the outcome of this sorting process.