Reducing energy consumption, increasing network throughput, and reducing delay are the pivot issues for wake-up radio- (WuR-) enabled wireless sensor networks (WSNs). In this paper, a relay selection joint consecutive packet routing (RS-CPR) scheme is proposed to reduce channel competition conflicts and energy consumption, increase network throughput, and then reduce end-to-end delay in data transmission for WuR-enabled WSNs. The main innovations of the RS-CPR scheme are as follows: (1) Relay selection: when selecting a relay node for routing, the sender will select the node with the highest evaluation weight from its forwarding node set (FNS). The weight of the node is weighted by the distance from the node to sink, the number of packets in the queue, and the residual energy of the node. (2) The node sends consecutive packets once it accesses the channel successfully, and it gives up the channel after sending all packets. Nodes that fail the competition sleep during the consecutive packet transmission of the winner to reduce collisions and energy consumption. (3) Every node sets two thresholds: the packet queue length threshold Nt and the packet maximum waiting time threshold Tt. When the corresponding value of the node is greater than the threshold, the node begins to contend for the channel. Besides, to make full use of energy and reduce delay, the threshold of nodes which are far from sink is small while that of nodes which are close to sink is large. In such a way, nodes in RS-CPR scheme will select those with much residual energy, a large number of packets, and a short distance from sink as relay nodes. As a result, the probability that a node with no packets to transmit becomes a relay is very small, and the probability that a node with many data packets in the queue becomes a relay is large. In this strategy, only a few nodes in routing need to contend for the channel to send packets, thereby reducing channel contention conflicts. Since the relay node has a large number of data packets, it can send many packets continuously after a successful competition. It also reduces the spending of channel competition and improves the network throughput. In summary, RS-CPR scheme combines the selection of relay nodes with consecutive packet routing strategy, which greatly improves the performance of the network. As is shown in our theoretical analysis and experimental results, compared with the receiver-initiated consecutive packet transmission WuR (RI-CPT-WuR) scheme and RI-WuR protocol, the RS-CPR scheme reduces end-to-end delay by 45.92% and 65.99%, respectively, and reduces channel collisions by 51.92% and 76.41%. Besides, it reduces energy consumption by 61.24% and 70.40%. At the same time, RS-CPR scheme improves network throughput by 47.37% and 75.02%.