•The basic reproduction number R0 for India was 2.08.•The Rt for India reduced to 1.16 on 22 April, about a month into the nation-wide lockdown.•A gradual lockdown relaxation farther after the first peak has passed is beneficial.•Testing and isolation prevent more secondary infections when social restrictions are relaxed.•Ramping up testing has many benefits and is much more economically feasible than extending lockdowns. India imposed one of the world’s strictest population-wide lockdowns on March 25, 2020 for COVID-19. We estimated epidemiological parameters, evaluated the effect of control measures on the epidemic in India, and explored strategies to exit lockdown. We obtained patient-level data to estimate the delay from onset to confirmation and the asymptomatic proportion. We estimated the basic and time-varying reproduction number (R0 and Rt) after adjusting for imported cases and delay to confirmation using incidence data from March 4 to April 25, 2020. Using a SEIR-QDPA model, we simulated lockdown relaxation scenarios and increased testing to evaluate lockdown exit strategies. R0 for India was estimated to be 2·08, and the Rt decreased from 1·67 on March 30 to 1·16 on April 22. We observed that the delay from the date of lockdown relaxation to the start of the second wave increases as lockdown is extended farther after the first wave peak—this delay is longer if lockdown is relaxed gradually. Aggressive measures such as lockdowns may be inherently enough to suppress an outbreak; however, other measures need to be scaled up as lockdowns are relaxed. Lower levels of social distancing when coupled with a testing ramp-up could achieve similar outbreak control as an aggressive social distancing regime where testing was not increased.