Home automation Internet of Things (IoT) systems have recently become a target for several types of attacks. In this paper, we present an authentication and key agreement protocol for a home automation network based on the ZigBee standard, which connects together a central controller and several end devices. Our scheme performs mutual authentication between end devices and the controller, which is followed by device-to-device communication. The scheme achieves confidentiality, message integrity, anonymity, unlinkability, forward and backward secrecy, and availability. Our scheme uses only simple hash and XOR computations and symmetric key encryption, and hence is resource-efficient. We show using a detailed security analysis and numerical results that our proposed scheme provides better security and anonymity, and is more efficient in terms of computation time, communication cost, and storage cost than schemes proposed in prior works.