An elastic nonwoven laminate that contains an elastic film laminated to one or more nonwoven facings is provided. The nonwoven facing contains a conventional polyolefin and can also contain a polyolefin-based plastomer. The laminate is activated by grooving to decouple the nonwoven facing from the elastic film. To reduce fiber-pull out that can result due to activation by grooving, the laminate can be post-bonded at room or elevated temperatures and a specific range of pressures to compact the fibers of the facing and minimize fiber pull-out/fuzziness while not sacrificing the softness, elasticity, and feel of the laminate.