Until recently, a dedicated mitotic apparatus that segregates newly replicated chromosomes into daughter cells was believed to be unique to eukaryotic cells. Here we demonstrate that the bacterium Caulobacter crescentus segregates its chromosome using a partitioning (Par) apparatus that has surprising similarities to eukaryotic spindles. We show that the C. crescentus ATPase ParA forms linear polymers in vitro and assembles into a narrow linear structure in vivo. The centromere-binding protein ParB binds to and destabilizes ParA structures in vitro. We propose that this ParB-stimulated ParA depolymerization activity moves the centromere to the opposite cell pole through a burnt bridge Brownian ratchet mechanism. Finally, we identify the pole-specific TipN protein as a new component of the Par system that is required to maintain the directionality of DNA transfer towards the new cell pole. Our results elucidate a bacterial chromosome segregation mechanism that features basic operating principles similar to eukaryotic mitotic machines, including a multivalent protein complex at the centromere that stimulates the dynamic disassembly of polymers to move chromosomes into daughter compartments.