We develop population models of high-mass X-ray binaries (HMXBs) formed after bursts of star formation and we investigate the effect of electron-capture supernovae (ECS) of massive ONeMg white dwarfs and the hypothesis that ECS events are associated with typically low supernova kicks imparted to the nascent neutron stars. We identify an interesting ECS bump in the time evolution of HMXB numbers; this bump is caused by significantly increased production of wind-fed HMXBs 20-60 Myr post-starburst. The amplitude and age extent of the ECS bump depend on the strength of ECS kicks and the mass range of ECS progenitors. We also find that ECS-HMXBs form through a specific evolutionary channel that is expected to lead to binaries with Be donors in wide orbits. These characteristics, along with their sensitivity to ECS properties, provide us with an intriguing opportunity to probe ECS physics and progenitors through studies of starbursts of different ages. Specifically, the case of the Small Magellanic Cloud, with a significant observed population of Be-HMXBs and starburst activity 30-60 Myr ago, arises as a promising laboratory for understanding the role of ECS in neutron star formation.