Current soft pneumatic grippers cannot robustly grasp flat materials and flexible objects on curved surfaces without distorting them. Current electroadhesive grippers, on the other hand, are difficult to actively deform to complex shapes to pick up free-form surfaces or objects. An easy-to-implement PneuEA gripper is proposed by the integration of an electroadhesive gripper and a two-fingered soft pneumatic gripper. The electroadhesive gripper was fabricated by segmenting a soft conductive silicon sheet into a two-part electrode design and embedding it in a soft dielectric elastomer. The two-fingered soft pneumatic gripper was manufactured using a standard soft lithography approach. This novel integration has combined the benefits of both the electroadhesive and soft pneumatic grippers. As a result, the proposed PneuEA gripper was not only able to pick-and-place flat and flexible materials such as a porous cloth but also delicate objects such as a light bulb. By combining two soft touch sensors with the electroadhesive, an intelligent and shape-adaptive PneuEA material handling system has been developed. This work is expected to widen the applications of both soft gripper and electroadhesion technologies.