In superconducting/ferromagnet heterostructures, spin-triplet Cooper pairs that carry spin information are crucial for the realization of superconducting spintronics. It has been theoretically proposed that they can be generated and controlled by the magnetic proximity effect and spin orbit coupling (SOC), resulting in a change in the critical temperature (T
C). However, experiments are still lacking, which limits the development of device applications. Here, we fabricate a series of Co/Nb/Pt heterostructures and reference samples, where the SOC and magnetic proximity can be independently controlled. We found the suppression of T
C up to 11% by the different Rashba-SOC at the Nb/Pt and Nb/Cu interfaces, and the normalized T
C/T
C
0 was successfully tuned by 24% with the coeffect of SOC and magnetic proximity. Through rigorous comparison, we attribute the tuning of superconductivity to the spin-triplet Cooper pairs, which is controlled by both the SOC strength and magnetic proximity. Our results offer a new pathway to control superconductivity with SOC and pave a new direction for the design of superconducting spintronics devices.