The phenomenon of storm surge and river interaction in estuarine systems is a multifaceted process affecting both dynamics and behaviour of water bodies. The present investigation aims to scrutinize the interaction between the surge and river flow during Cyclone Amphan in May 2020 in the Hooghly Estuary. The influence of river discharge on the propagation of tidal waves during the event is also investigated. To this end, a shock-capturing finite-volume based in-house model IROMS-C2D is employed to simulate the surge-river dynamics during the cyclone event. The model solves the governing equations of shallow water equations for the river flow with additional forcing terms such as wind pressure and stress and Coriolis for surge propagation. Unlike the commonly adopted loosely coupled models for the two processes, IROMS-C2D simulates the river-surge flow interaction in an integral way. The model is set up for the Hooghly River from the Open Ocean to Triveni of the Bay of Bengal. The computational domain is defined by triangular grids with river and ocean bathymetry. The inflow at the upstream of Hooghly River, water level variation at the ocean boundary, and the Amphan track with its parameters are given as inputs to the model. The analysis reveals that the surge-river interaction in the Hooghly estuary during the Amphan cyclone was marked by a convoluted interplay between surge, river discharge, and tidal dynamics. Specifically, the discharge along the river significantly impacted the surge penetration along the river, with higher discharge rates leading to reduced surge penetration and vice versa. This behaviour can be attributed to the combined effect of river discharge and tidal dynamics, which regulate the magnitude and direction of the flow in the estuary. The model mimics the propagation of tidal waves and provides estimations of the velocity magnitude during a cyclone despite using a depth-averaged velocity profile. The research suggests that the developed model is a valuable and cost-effective tool for describing the surge-river interaction, coastal inundation, and complex flow dynamics in the estuary during a cyclonic event and could be used to support field observation. The present study contributes to a comprehensive understanding of the mechanisms underlying surge-river interaction in estuarine systems during extreme weather conditions. The findings have substantial implications for predicting and mitigating storm surge impacts in the Hooghly Estuary and other similar coastal-riverine systems. The outcome of the study can inform the development of more robust and effective early warning systems and disaster management strategies that safeguard people and coastal infrastructure during severe weather events.