We study the asymptotic distribution of the number of fiber intersections in two-dimensional random fiber webs. Previous work on such systems has focused primarily on first- and second-order moment characteristics derived from geometric probability models. In this paper, we move beyond moment-based analysis and establish asymptotic normality for normalized intersection counts. For a planar fiber web consisting of line segments randomly placed with uniform locations and orientations, we show that the appropriately normalized number of intersections within a bounded observation window converges in distribution to a Gaussian limit as the number of fibers diverges. We first derive a central limit theorem for the equal-length fiber model and then extend the result to heterogeneous fiber systems with a mixture of two fiber lengths. In both cases, explicit expressions for the asymptotic mean and variance are obtained in terms of the first two moments of the truncated fiber lengths within the counting region. Theoretical results are supported by Monte Carlo simulations, which demonstrate that the Gaussian approximation provides an accurate description of the finite-sample distribution of the intersection count. Our findings place random fiber intersection models within the broader framework of stochastic geometry and provide a foundation for statistical inference in fiber-based network systems.