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Cuprate high-temperature superconductors feature a complex phase diagram where various forms of symmetry breaking are found in close proximity to superconductivity (SC). Recent ubiquitous observations of charge density wave order (CDW) across families of cuprate superconductors suggest CDW a genuine instability of the doped CuO₂ plane. Here we report resonant soft x-ray scattering (RSXS) study of CDW in La₂CuO_4+y, which features 3-dimensional ordering of oxygen dopants, and simultaneous onset of SC and spin density wave (SDW) at T_c ~ 42 K. Emerging first well above T_c, the CDW in La₂CuO_4+y is found to be dramatically enhanced below T_c. This contrast with previous observations in other cuprate superconductors where the CDW is suppressed by the onset of SC. Our results on La₂CuO_4+y provide evidence of cooperative relationship between CDW, SDW, and SC in a clean cuprate superconductor, and further corroborate the view point of "intertwined orders" to describe the complex electronic orderings in strongly correlated materials.