The invention relates to an elastic damping structure and a connection method. The damping structure comprises a fastener and a damping assembly; the damping assembly connects an upper mounting plate with a lower mounting plate together through the fastener; the lower mounting plate is provided with a mounting hole; the damping assembly comprises an upper elastic damping cushion, a lower elastic damping cushion and an elastic damping cylinder; the upper and lower elastic damping cushions are provided with through holes with the same sizes correspondingly; the outer diameter of the elastic damping cylinder is matched with the inner diameters of the through holes of the upper and lower elastic damping cushions; the elastic damping cylinder penetrates through the mounting hole of the lower mounting plate; the elastic damping cylinder is sleeved with the upper and lower elastic damping cushions; and the upper and lower elastic damping cushions are located on the upper and lower sides of the lower mounting plate correspondingly. The elastic damping structure is simple in structure, easy to machine and manufacture and easy to assemble, at the time of using, mounting is convenient, the damping effect is good, and universality is high.