Testing capacity for COVID-19 remains a challenge globally due to the lack of adequate supplies, trained personnel, and sample-processing equipment. These problems are even more acute in rural and underdeveloped regions. We demonstrate that solicited-cough sounds collected over a phone, when analysed by our AI model, have statistically significant signal indicative of COVID-19 status (AUC 0.72, t-test,p <0.01,95% CI 0.61-0.83). This holds true for asymptomatic patients as well. Towards this, we collect the largest known(to date) dataset of microbiologically confirmed COVID-19 cough sounds from 3,621 individuals. When used in a triaging step within an overall testing protocol, by enabling risk-stratification of individuals before confirmatory tests, our tool can increase the testing capacity of a healthcare system by 43% at disease prevalence of 5%, without additional supplies, trained personnel, or physical infrastructure