We investigate the issue of $p$-th moment exponentially input-to-state stability $(p\operatorname{MEISS})$ of multi-group stochastic coupled systems with time-varying delay (MSCSTD) in this paper. By means of results from graph theory, we develop a systematic method that allows one to construct a proper Lyapunov function for MSCSTD. More specifically, two kinds of sufficient criteria, which are called Lyapunov-type and coefficient-type respectively, are derived to ensure $p\operatorname{MEISS}$ for MSCSTD by using the graph-theoretic approach. To make results more understandable, we apply them to a typical stochastic coupled oscillators with control inputs.