Autism Spectrum Disorder (ASD) affects 1 in 54 children in the United States. A core social communication skill negatively impacted by ASD is joint attention (JA), which influences the development of language, cognitive, and social skills from infancy onward. Although several technology-based JA studies have shown potential, they primarily focus on response to joint attention (RJA). The other important component of JA, the initiation of joint attention (IJA), has received less attention from a technology-based intervention perspective. In this work, we present an immersive Computer-mediated Caregiver-Child Interaction (C3I) system to help children with ASD practice IJA skills. C3I is a novel computerized intervention system that integrates a caregiver in the teaching loop, thereby preserving the advantages of both human and computer-administered intervention. A feasibility study with 6 dyads (caregiver-child with ASD) was conducted. A near significant increase with medium effect size on IJA performance was observed. Meanwhile, physiology-based stress analysis showed that C3I did not increase stress of the caregivers over the course of the study. To the best of our knowledge, this is the first autonomous system designed for teaching IJA skills to children with ASD incorporating caregivers within the loop to enhance the potential for generalization in real-world.