We show that a collinear Heisenberg antiferromagnet, whose sublattice symmetry is broken at the Hamiltonian level, becomes a fluctuation-induced ferrimagnet at any finite temperature T below the Néel temperature TN. We demonstrate this using a layered variant of a square-lattice J1-J2 model. Linear spin-wave theory is used to determine the low-temperature behavior of the uniform magnetization, and nonlinear corrections are argued to yield a temperature-induced qualitative change of the magnon spectrum. We then consider a layered Shastry-Sutherland model, describing a frustrated arrangement of orthogonal dimers. This model displays an antiferromagnetic phase for large intradimer couplings. A lattice distortion which breaks the glide symmetry between the two types of dimers corresponds to broken sublattice symmetry and hence gives rise to ferrimagnetism. Given indications that such a distortion is present in the material SrCu2(BO3)2 under hydrostatic pressure, we suggest the existence of a fluctuation-induced ferrimagnetic phase in pressurized SrCu2(BO3)2. We predict a nonmonotonic behavior of the uniform magnetization as function of temperature.