Identifying the causes for historical sea-level changes in coastal tide-gauge records is important for constraining oceanographic, geologic, and climatic processes. The Río de la Plata estuary in South America features the longest tide-gauge records in the South Atlantic. Despite the relevance of these data for large-scale circulation and climate studies, the mechanisms underlying relative sea-level changes in this region during the past century have not been firmly established. I study annual data from tide gauges in the Río de la Plata and stream gauges along the Río Paraná and Río Uruguay to establish relationships between river streamflow and sea level over 1931-2014. Regression analysis suggests that streamflow explains 59%+/-17% of the total sea-level variance at Buenos Aires, Argentina, and 28%+/-21% at Montevideo, Uruguay (95% confidence intervals). A longterm streamflow increase effected sea-level trends of 0.71+/-0.35 mm/yr at Buenos Aires and 0.48+/-0.38 mm/yr at Montevideo. More generally, sea level at Buenos Aires and Montevideo respectively rises by (7.3+/-1.8)x10^-6 m and (4.7+/-2.6)x10^-6 m per 1 m^3 s^-1 streamflow increase. These observational results are consistent with simple theories for the coastal sea-level response to streamflow forcing, suggesting a causal relationship between streamflow and sea level mediated by ocean dynamics. Findings advance understanding of local, regional, and global sea-level changes, clarify sea-level physics, inform future projections of coastal sea level and the interpretation of satellite data and proxy reconstructions, and highlight future research directions.