We carried out Chandra observations of two low-luminosity low-mass X-ray binaries, 1RXS J170854.4-321857 and 1RXS J171824.2-402934, for which previously single X-ray bursts were detected with the Wide Field Cameras (WFCs) on board BeppoSAX. Both were detected in our Chandra observations in an actively accreting state three to eight years after the X-ray bursts, with 0.5-10 keV luminosities between 5×1034 and 2×1036 erg s-1. The apparently persistent nature is remarkable for 1RXS J171824.2-402934 given its low luminosity of 10-3 L_Edd. The persistence of both sources also distinguishes them from 5 other low-L bursters, which have also been seen during bursts with the WFCs but were not detected during Chandra observations above a luminosity of 1033 erg s-1. Those are probably transient rather than persistent sources.