Anthropogenic forcing is increasing the likelihood and severity of certain extreme weather events, which may catalyze outbreaks of climate-sensitive infectious diseases. Extreme precipitation events can promote the spread of mosquito-borne illnesses by creating vector habitat, destroying infrastructure, and impeding vector control. Here, we focus on Cyclone Yaku, which caused heavy rainfall in northwestern Peru from March 7th - 20th, 2023 and was followed by the worst dengue outbreak in Peru's history. We apply generalized synthetic control methods to account for baseline climate variation and unobserved confounders when estimating the causal effect of Cyclone Yaku on dengue cases across the 56 districts with the greatest precipitation anomalies. We estimate that 67 (95% CI: 30 - 87)% of cases in cyclone-affected districts were attributable to Cyclone Yaku. The cyclone significantly increased cases for over six months, causing 38,209 (95% CI: 17,454 - 49,928) out of 57,246 cases. The largest increases in dengue incidence due to Cyclone Yaku occurred in districts with a large share of low-quality roofs and walls in residences, greater flood risk, and warmer temperatures above 24°C. Analyzing an ensemble of climate model simulations, we found that extremely intense March precipitation in northwestern Peru is 42% more likely in the current era compared to a preindustrial baseline due to climate forcing. In sum, extreme precipitation like that associated with Cyclone Yaku has become more likely with climate change, and Cyclone Yaku caused the majority of dengue cases across the cyclone-affected districts.