One efficient mechanism for generating a charge supercurrent is Andreev reflection, in which the electric current injected from a normal metal into a conventional superconductor is converted into a supercurrent, thereby preserving charge conservation. We here propose a general principle for generating spin supercurrents in unitary (equal-spin) triplet superconductors by analogy with such charge transport, i.e., assuming spin conservation. We find a spin torque that is proportional to the triplet superconducting order parameter and, in the spin-conservation scenario, converts the particle spin to that of Cooper pairs. Based on this general principle, we propose an implementation to efficiently generate a spin supercurrent in unitary triplet superconductors, even though the condensate carries no spin polarization at equilibrium, by the magnetization dynamics ${\bf M}(t)$ of a proximity magnetic nanostructure. In this process, the quasiparticle spin current is also efficiently generated. The efficiency of spin pumping of both Cooper pairs and quasiparticles is not solely limited to the $d{\bf M}/dt\times {\bf M}$ due to the emergent particle-hole symmetry, thereby going beyond the conventional spin pumping of electrons. This general principle provides an efficient approach to generating and manipulating dissipationless spin currents in many unconventional superconductors.