Everyday experience suggests that people are equally aware of stimuli in both hemifields. However, when two streams of stimuli are rapidly presented left and right, the second target (T2) is better identified in the left hemifield than in the right hemifield. This left visual field (LVF) advantage may result from differences between hemifields in attracting attention. Therefore, we introduced a visual cue shortly before T2 onset to draw attention to one stream. Thus, to identify T2, attention was correctly positioned with valid cues but had to be redirected to the other stream with invalid ones. If the LVF advantage is caused by differences between hemifields in attracting attention, invalid cues should increase, and valid cues should reduce the LVF advantage as compared with neutral cues. This prediction was confirmed. ERP analysis revealed that cues evoked an early posterior negativity, confirming that attention was attracted by the cue. This negativity was earlier with cues in the LVF, which suggests that responses to salient events are faster in the right hemisphere than in the left hemisphere. Valid cues speeded up, and invalid cues delayed T2-evoked N2pc; in addition, valid cues enlarged T2-evoked P3. After N2pc, right-side T2 evoked more sustained contralateral negativity than left T2, least long-lasting after valid cues. Difficulties in identifying invalidly cued right T2 were reflected in prematurely ending P3 waveforms. Overall, these data provide evidence that the LVF advantage is because of different abilities of the hemispheres in shifting attention to relevant events in their contralateral hemifield.