The COBRA collaboration searches for neutrinoless double beta-decay (0νββ-decay) using CdZnTe semiconductor detectors with a coplanar-grid readout and a surrounding guard-ring structure. The operation of the COBRA demonstrator at the Gran Sasso underground laboratory (LNGS) indicates that alpha-induced lateral surface events are the dominant source of background events. By instrumenting the guard-ring electrode it is possible to suppress this type of background. In laboratory measurements this method achieved a suppression factor of alpha-induced lateral surface events of 5300 +2660 −1380 , while retaining (85.3 ± 0.1) % of gamma events occurring in the entire detector volume. This suppression is superior to the pulse-shape analysis methods used so far in COBRA by three orders of magnitude.