An absorbent article containing a nonwoven web that includes a plurality of polyolefin fibers is provided. The polyolefin fibers are formed by a thermoplastic composition containing a continuous phase that includes a polyolefin matrix polymer and nanoinclusion additive is provided. The nanoinclusion additive is dispersed within the continuous phase as discrete nano-scale phase domains. When drawn, the nano-scale phase domains are able to interact with the matrix in a unique manner to create a network of nanopores.