This paper presents a new approximation method for pricing multi-asset continuous
single barrier options under general local stochastic volatility models. The formula applies an asymptotic expansion technique and an approximation for the distribution of the first exit time of diffusion processes. This method focuses on local stochastic volatility models with unknown characteristic function and transition density function. To the best of our knowledge, our approximation formula is the first to achieve analytic approximations for continuous barrier options prices in this environment. In numerical experiments, we confirm the validity of the formula.