Catastrophic phenomena such as earthquakes, avalanches, and critical points in the stock market are hard to predict. Recently, Xia et al. [Pure Appl. Geophys. 159 (2002)2491] showed that a geometrical system near catastrophic rupture presents a general critical sensitivity: the system becomes significantly sensitive near the catastrophe transition. Here we report that the phenomenon of critical sensitivity also exists in small-world network systems. With the increase of the small-world rewiring probability p, from the regular network p = 0 to the random one p = 1, the system performs more sensitively before the critical point while remaining in better organization through the evolutional progress, and the prediction threshold Ps performs more in advance. The concept of critical sensitivity can be applied to other complex network systems.