Bacteriophages (phages), viruses that exclusively infect bacterial hosts, are the most abundant living entities across diverse environments, coexisting with their bacterial hosts at densities exceeding 10^7 ml-1 in marine surface waters, 108 ml-1 in soils, and 109 ml-1 in the human gut. In contrast, virulent phage rapidly lyse bacteria populations within well-mixed in vitro environments, selecting for the emergence of phage-resistant bacterial mutants, which in turn select for host-range expansion phage mutants, leading to the emergence of complex cross-infection networks, and eventually the collapse of phage populations altogether. This gap in outcome raises a question: what enables long-term phage-bacteria coexistence? Here, we show how interactions in space can facilitate stable coexistence and long-range transport of virulent phage along with migrating bacteria. Through the joint use of theory, simulation, and experiments across multiple phage-bacteria systems, we reveal a chemotaxis- driven mechanism which robustly stabilizes coexistence and dispersal of virulent phages with migrating hosts, while minimizing the potential for coevolutionary-induced collapse of either bacteria or phage. These findings suggest the ecological relevance of spatial interaction mechanisms that reinforce stability between antagonistic partners, even in the absence of perpetual cycles of defense and counter-defense, that may be broadly applicable across phage-bacteria pairs.