This paper studies the asymptotic tail behaviour of products of independent Poisson random variables. Let \[ Z_m=\prod_{j=1}^m X_j, \] where $X_1,\dots,X_m$ are independent Poisson random variables. We derive a Laplace-type asymptotic approximation for \[ P(Z_m \ge n), \qquad n\to\infty, \] whose relative error tends to zero. The analysis is based on Stirling's logarithmic approximation, a constrained saddle-point method, the Lambert $W$ function, and a careful evaluation of the associated Gaussian prefactor. These tools yield an explicit asymptotic description of the tail probability of the product. For clarity of exposition, we first treat the case $m=2$, which illustrates the main ideas in a simpler setting, and then extend the argument to the general product of $m$ independent Poisson random variables.