Investigation of interaction between solitons and their background small amplitude waves has been an interesting topic in numerical study for more than three decades. A classical soliton accompanied with oscillatory tails to infinite extent in space, is an interesting quasi-soliton, which has been revealed in experimental study and really observed. However, analytical solution of such a special quasi-soliton structure is rarely considered. In this paper, two branches of soliton-cnoidal wave solution as well as the two-soliton solution of the Korteweg-de Vries (KdV) equation are obtained by the generalized tanh expansion method. The exact relation between the soliton-cnoidal wave solution and the classical soliton solution of the KdV equation is established. By choosing suitable wave parameters, the quasi-soliton behavior of the soliton-cnoidal wave solution is revealed. It is found that with modulus of the Jacobi elliptic function approaching to zero asymptotically, the oscillating tails can be minimized and the soliton core of the soliton-cnoidal wave turns closer to the classical soliton solution. In addition, the quasi-soliton structure is revealed in a plasma physics system. By the reductive perturbation approach, the KdV equation modeling ion acoustic waves in an ideal homogeneous magnetized plasma is derived. It is confirmed that the waveform of the quasi-soliton is significantly influenced by the electron distribution, temperature ratio of ion to electron, magnetic field strength, and magnetic field direction. Interestingly, the amplitude of the quasi-soliton keeps constant due to the -independence of nonlinear coefficient A. The width of the soliton core and the wavelength of the surrounded periodic wave become constant with the further increase of . The explicit soliton-cnoidal wave solution with quasi-soliton behavior obtained here is applicable to many physical scenarios. For instance, the quasi-soliton structure can be viewed as a classical soliton with perturbations, and can correct the classical soliton in both theoretical and experimental study.