One of the significant challenges in the Internet of Things (IoT) is the provisioning of guaranteed security and privacy, considering the fact that IoT devices are resource-limited. Oftentimes, in IoT applications, remote users need to obtain real-time data, with guaranteed security and privacy, from resource-limited network nodes through the public Internet. For this purpose, the users need to establish a secure link with the network nodes. Though the IPv6 over low-power wireless personal area networks (6LoWPAN) adaptation layer standard offers IPv6 compatibility for resource-limited wireless networks, the fundamental 6LoWPAN structure ignores security and privacy characteristics. Thus, there is a pressing need to design a resource-efficient authenticated key exchange (AKE) scheme for ensuring secure communication in 6LoWPAN-based resource-limited networks. This paper proposes a resource-efficient secure remote user authentication scheme for 6LoWPAN-based IoT networks, called SRUA-IoT. SRUA-IoT achieves the authentication of remote users and enables the users and network entities to establish private session keys between themselves for indecipherable communication. To this end, SRUA-IoT uses a secure hash algorithm, exclusive-OR operation, and symmetric encryption primitive. We prove through informal security analysis that SRUA-IoT is secured against a variety of malicious attacks. We also prove the security strength of SRUA-IoT through formal security analysis conducted by employing the random oracle model. Additionally, we prove through Scyther-based validation that SRUA-IoT is resilient against various attacks. Likewise, we demonstrate that SRUA-IoT reduces the computational cost of the nodes and communication overheads of the network.