Despite growing interest in quantifying and modeling the scoring dynamics within professional sports games, relative little is known about what patterns or principles, if any, cut across different sports. Using a comprehensive data set of scoring events in nearly a dozen consecutive seasons of college and professional (American) football, professional hockey, and professional basketball, we identify several common patterns in scoring dynamics. Across these sports, scoring tempo---when scoring events occur---closely follows a common Poisson process, with a sport-specific rate. Similarly, scoring balance---how often a team wins an event---follows a common Bernoulli process, with a parameter that effectively varies with the size of the lead. Combining these processes within a generative model of gameplay, we find they both reproduce the observed dynamics in all four sports and accurately predict game outcomes. These results demonstrate common dynamical patterns underlying within-game scoring dynamics across professional team sports, and suggest specific mechanisms for driving them. We close with a brief discussion of the implications of our results for several popular hypotheses about sports dynamics.