In order to overcome the deficiencies of large water consumption and insufficient damping performance of the traditional tuned liquid damper (TLD), a novel tuned liquid inerter damper by combining a rack and pinion inerter device with a damping net and sloped-bottom tuned liquid damper (DNS-TLD) which is named as damping net and sloped-bottom tuned liquid inerter damper (DNS-TLID) is proposed. The damping mechanism of DNS-TLID is analyzed and the solution formula about fundamental frequency is derived, then a mechanical model of the DNS-TLID-single-degree-of-freedom (SDOF) structural system is established. A functional solution method for estimating equivalent liquid motion velocity is presented, and a simplified calculation method of the total equivalent damping ratio of DNS-TLID is proposed. The accuracy of the proposed mechanical model and the calculation formula for the total equivalent damping ratio is verified through the shaking table tests of one single-story steel frame structure. The results from both theoretical analysis and experimental investigation demonstrate that DNS-TLID can effectively achieve lightweight vibration control and significantly enhance the damping performance of structures subjected to ground motions. Compared to other types of TLDs, DNS-TLID can effectively integrate the advantages of DNS-TLD and the rack and pinion inerter device and exhibit superior performance and lightweight control effects. Thus, as a novel advanced combination damper, DNS-TLID proves to be highly applicable in engineering applications.