The quantification of system reliability is fundamental to the assessment of a system’s safety and resilience, and has been of interest to decision-makers. Since quantifying the system reliability is shown to be computationally intractable, researchers aim to find ap- proximations. Existing approaches to approximate reliability either suffer from poor scal- ability or lack of correctness guarantees. Answer Set Programming (ASP) is a powerful tool for knowledge representation that can specify complex combinatorial problems. In recent years, the new applications of ASP have propelled the emergence of well-engineered ASP systems. This paper proposes a new ASP counting based framework, RelNet-ASP, to approximate or estimate the reliability of a system or network. The framework reduces the problem of reliability estimation to an approximate model counting problem on ASP programs, offering formal guarantees of the estimated reliability. The experimental eval- uation demonstrates that RelNet-ASP outperforms state-of-the-art techniques in terms of both runtime performance and accuracy.