Sub-synchronous oscillations (SSOs) originating from inverter-based resources (IBRs), which can be specifically induced by different controls within IBRs, have become a significant challenge to the stability of power system. Identifying the specific control responsible for the oscillation is crucial for effective oscillation mitigation. However, the coupling of multi-frequency disturbance terms, along with the complexities introduced by nonlinear elements, makes it difficult to isolate the impact of individual control. To address this issue, this paper proposes a control-level SSO source identification method for IBRs-dominated power systems based on initial instantaneous power characteristics. First, the expressions of instantaneous power during SSOs induced by different controls are derived for grid-following (GFL)-IBRs and grid-forming (GFM)-IBRs. The causes of current saturation triggering are revealed, and the coupling effects introduced by nonlinear elements are examined. Then, the initial instantaneous power characteristics of grid-side converter (GSC)-control-induced oscillations (GCIOs) and synchronization-control-induced oscillations (SCIOs) are analyzed, by comparing cases where the saturation limit is reached and not reached. The proposed method incorporates the dynamic characteristics of IBR controls, and can identify the SSO source using measurement data at the IBR port. Finally, the effectiveness of the proposed method in various SSO scenarios is validated through real-time simulations. Comparisons with existing methods demonstrate that the proposed method achieves higher identification accuracy.