Disclosed is a melt-blown fiber web with improved concentration force and elasticity, whereby a melt-blown fabric is cut and sealed at predetermined intervals using knives having arbitrary patterns so that concentration force and elasticity of the melt-blown fiber web can be improved without degrading the inherent function of the fiber web. Further disclosed are a method and apparatus for manufacturing the melt-blown fiber web. The melt-blown fiber web includes thermoplastic filaments, wherein cutting portions and sealing portions are arranged on top and bottom surfaces of the fiber web at predetermined intervals along a thickness of the fiber web so that a concentration force and elasticity of the fiber web are improved.