Continual learning is crucial for creating AI agents that can learn and improve themselves autonomously. A primary challenge in continual learning is to learn new tasks without losing previously learned knowledge. Current continual learning methods primarily focus on enabling a neural network with mechanisms that mitigate forgetting effects. Inspired by the two distinct systems in the human brain, System 1 and System 2, we propose a Neuro-Symbolic Brain-Inspired Continual Learning (NeSyBiCL) framework that incorporates two subsystems to solve continual learning: A neural network model responsible for quickly adapting to the most recent task, together with a symbolic reasoner responsible for retaining previously acquired knowledge from previous tasks. Moreover, we design an integration mechanism between these components to facilitate knowledge transfer from the symbolic reasoner to the neural network. We also introduce two compositional continual learning benchmarks and demonstrate that NeSyBiCL is effective and leads to superior performance compared to continual learning methods that merely rely on neural architectures to address forgetting.