To study the displacement response of pile foundations, the dynamic response of pile-soil interface, and the law of stress distribution in the soil around the pile when the top of the pile is subjected to horizontal transient loads, such as impact, under the condition of continuity of the pile-soil interface, the simplified model of pile-saturated soil coupling vibration is established using the Euler beam based on the Biot theory and Novak plane strain assumption. The system’s dynamic control equations are solved using Laplace transformation and potential function decomposition. The focus is on the time-domain analysis of the vibration response of the pile-soil system under horizontal triangular impact loading, including pile displacement response, pile-soil interface, and dynamic response of the soil around the pile. The research reveals that the displacement field response of the pile-soil system lags behind the stress field response. Moreover, as the pile-soil modulus ratio decreases, the effective radial stress, shear stress, and pore pressure responses at the pile-soil interface become more pronounced. A decrease in soil permeability coefficient leads to increasing pore pressure at the pile-soil interface, causing a reduction of effective radial stress. The shear stress at the pile-soil interface remains almost unchanged with varying permeability coefficients. For higher permeability coefficients, the pore pressure distribution around the pile is more dispersed, and the maximum effective radial stress occurs near the pile-soil interface. Conversely, for lower permeability coefficients, the pore pressure distribution around the pile is more concentrated, and the maximum effective radial stress occurs at a greater distance from the pile-soil interface.