Countries in the Indian subcontinent have committed to reducing the incidence of kala-azar, a clinical manifestation of visceral leishmaniasis, to below 1 in 10,000 by 2020. We address the role of timing of use and accuracy of diagnostics in kala-azar control and elimination. We use empirical data on health-seeking behaviour and health-system performance from the Indian state of Bihar, Bangladesh and Nepal to parameterize a mathematical model. Diagnosis of cases is key to case management, control and surveillance. Treatment of cases prevents onward transmission, and we show that the differences in time to diagnosis in these three settings explain the observed differences in incidence. Shortening the time from health-care seeking to diagnosis is likely to lead to dramatic reductions in incidence in Bihar, bringing the incidence down to the levels seen in Bangladesh and Nepal. The results emphasize the importance of maintaining population and health-system awareness, particularly as transmission and disease incidence decline. We explore the possibility of diagnosing patients before the onset of clinical kala-azar (before 14 days fever), and show that this could have a marked impact on incidence, even for a moderately sensitive test. However, limited specificity (that results in false positives) is a major barrier to such a strategy. Diagnostic tests of high specificity used at an early stage of active infection, even if sensitivity is only moderate, could have a key role in the control of kala-azar, and prevent its resurgence when paired with the passive health-care system and tests of high sensitivity, such as the test for rK39 antibody response.