Accretion disks around neutron stars regularly undergo sudden strong irradiation by Type-I X-ray bursts powered by unstable thermonuclear burning on the stellar surface. We investigate the impact on the disk during one of the first X-ray burst observations with the Neutron Star Interior Composition Explorer (NICER) on the International Space Station. The burst is seen from Aql X-1 during the hard spectral state. In addition to thermal emission from the neutron star, the burst spectrum exhibits an excess of soft X-ray photons below 1 keV, where NICER's sensitivity peaks. We interpret the excess as a combination of reprocessing by the strongly photoionized disk and enhancement of the pre-burst persistent flux, possibly due to Poynting-Robertson drag or coronal reprocessing. This is the first such detection for a short sub-Eddington burst. As these bursts are observed frequently, NICER will be able to study how X-ray bursts affect the disk and corona for a range of accreting neutron star systems and disk states.