In this paper, we propose a new high-resolution shock-capturing scheme based on dimension-bydimension reconstruction. This scheme is intended for adaptively-refined Cartesian-based unstructured grids, and the computational cost is smaller than conventional full reconstruction schemes. In the dimension-by-dimension reconstruction, two reconstructed functions are used, and the cell-interface value is interpolated on these functions. The scheme achieves formal accuracy when combined with a new efficient flux quadrature scheme. Then the WENO procedure is introduced in the approximation of derivatives in each cell, and this prevent the derivatives from blowing up when discontinuity exists within the stencil. Finally, the present scheme is implemented in shock-capturing problems on uniform and adaptively-refined grids. The results shows better resolution than conventional MUSCL-type TVD schemes.