Background Leptospirosis is a bacterial zoonosis. Leptospirosis incidence (LI) in Sri Lanka is high. Infected animals excrete leptospires into the environment via their urine. Survival of leptospires in the environment until they enter into a person and several other factors that influence leptospirosis transmission are dependent upon local weather. Past studies show that rainfall and other weather parameters are correlated with the LI in the Kandy district, Sri Lanka. El Niño Southern Oscillation (ENSO), ENSO Modoki, and the Indian Ocean Dipole (IOD) are teleconnections known to be modulating rainfall in Sri Lanka. There is a severe dearth of published studies on the correlations between indices of these teleconnections and LI. Methods We acquired the counts of leptospirosis cases notified and midyear estimated population data of the Kandy district from 2004 to 2019, respectively, from weekly epidemiology reports of the Ministry of Health and Department of Census and Statistics of Sri Lanka. We estimated weekly and monthly LI of Kandy. We obtained weekly and monthly teleconnection indices data for the same period from the National Oceanic and Atmospheric Administration (NOAA) of the USA and Japan Agency for Marine-Earth Science and Technology (JAMSTEC). We performed wavelet time series analysis to determine correlations with lag periods between teleconnection indices and LI time series. Then, we did time-lagged detrended cross-correlation analysis (DCCA) to verify wavelet analysis results and to find the magnitudes of the correlations detected. Results Wavelet analysis displayed indices of ENSO, IOD, and ENSO Modoki were correlated with the LI of Kandy with 1.9–11.5-month lags. Indices of ENSO showed two correlation patterns with Kandy LI. Time-lagged DCCA results show all indices of the three teleconnections studied were significantly correlated with the LI of Kandy with 2–5-month lag periods. Conclusions Results of the two analysis methods generally agree indicating that ENSO and IOD modulate LI in Kandy by modulating local rainfall and probably other weather parameters. We recommend further studies about the ENSO Modoki and LI correlation in Sri Lanka. Monitoring for extreme teleconnection events and enhancing preventive measures during lag periods can blunt LI peaks that may follow.