Continual learning (CL) in the brain is facilitated by a complex set of mechanisms. This includes the interplay of multiple memory systems for consolidating information as posited by the complementary learning systems (CLS) theory and synaptic consolidation for protecting the acquired knowledge from erasure. Thus, we propose a general CL method that creates a synergy between SYNaptic consolidation and dual memory Experience Replay (SYNERgy). Our method maintains a semantic memory that accumulates and consolidates information across the tasks and interacts with the episodic memory for effective replay. It further employs synaptic consolidation by tracking the importance of parameters during the training trajectory and anchoring them to the consolidated parameters in the semantic memory. To the best of our knowledge, our study is the first to employ dual memory experience replay in conjunction with synaptic consolidation that is suitable for general CL whereby the network does not utilize task boundaries or task labels during training or inference. Our evaluation on various challenging CL scenarios and characteristics analyses demonstrate the efficacy of incorporating both synaptic consolidation and CLS theory in enabling effective CL in DNNs.