Reliability analysis of corroded pipelines is critical to the integrity and safe working of pipeline infrastructure. Aiming at less probability information is obtained for corrosion pipeline engineering, and the mechanical properties of pipeline with corrosion defects deteriorate caused by the accumulative effect of corrosion growth. Based on the quasi-static analysis method and non-probability theory, this paper presents a reliability model for assessing corroded pipelines with corrosion growth. In fact, reliability analysis of corroded pipelines needs to consider the interaction of multiple uncertainty variables. By introducing interaction theory, a mathematical model of corrosion defects considering the interaction of variables is put forward. Moreover, this paper develops a non-probabilistic time-varying reliability method for pipeline systems with multiple defects. Thus, several numerical examples are investigated to discuss the effectiveness of the proposed methodology. The results show that a two-dimensional or even three-dimensional ellipsoid model with correlation has more accurate results to evaluate corroded pipelines under the interaction of multiple corroded defects with poor information. Furthermore, a non-probabilistic time-varying reliability model is established according to the time-varying characteristics of the corroded pipeline under the influence of multiple factors. An effective complement to the theory of non-probabilistic reliability analysis of system is investigated. The analysis of the results suggests that interaction of corroded pipeline has a negligible impact on reliability. It also provides a theoretical basis for maintenance and is of great significance for risk- and reliability-informed decisions regarding buried oil and gas pipelines.