The evolution of complex organismal traits is obvious as a historical fact, but the underlying causes--including the role of natural selection--are contested. Gould argued that a random walk from a necessarily simple beginning would produce the appearance of increasing complexity over time. Others contend that selection, including coevolutionary arms races, can systematically push organisms toward more complex traits. Methodological challenges have largely precluded experimental tests of these hypotheses. Using the Avida platform for digital evolution, we show that coevolution of hosts and parasites greatly increases organismal complexity relative to that otherwise achieved. As parasites evolve to counter the rise of resistant hosts, parasite populations retain a genetic record of past coevolutionary states. As a consequence, hosts differentially escape by performing progressively more complex functions. We show that coevolution's unique feedback between host and parasite frequencies is a key process in the evolution of complexity. Strikingly, the hosts evolve genomes that are also more phenotypically evolvable, similar to the phenomenon of contingency loci observed in bacterial pathogens. Because coevolution is ubiquitous in nature, our results support a general model whereby antagonistic interactions and natural selection together favor both increased complexity and evolvability.