We propose a joint feature compression and transmission scheme for efficient inference at the wireless network edge. Our goal is to enable efficient and reliable inference at the edge server assuming limited computational resources at the edge device. Previous work focused mainly on feature compression, ignoring the computational cost of channel coding. We incorporate the recently proposed deep joint source-channel coding (DeepJSCC) scheme, and combine it with novel filter pruning strategies aimed at reducing the redundant complexity from neural networks. We evaluate our approach on a classification task, and show improved results in both end-to-end reliability and workload reduction at the edge device. This is the first work that combines DeepJSCC with network pruning, and applies it to image classification over the wireless edge.