High-efficiency near-UV excitable inorganic phosphors with tunable emissions are highly demanded for white light-emitting diodes (LEDs) and displays. In this paper, a series of color-tunable cyan-green-emitting Ca2Lu0.99-xZr2(AlO4)3:1%Ce3+,xTb3+ (CLZA:1%Ce3+,xTb3+) phosphors was synthesized via a high-temperature solid-state reaction method. Their crystal structure, morphology, emission and excitation spectra, energy transfer properties, CIE color coordinates, internal quantum efficiency and temperature-dependent spectra were systematically researched. Impressively, these CLZA:1%Ce3+,xTb3+ phosphors showed an intense excitation band in the 370–475 nm wavelength range with a peak at 408 nm corresponding to the 4f→5d transitions of Ce3+ ions. Owing to the energy transfer from Ce3+ to Tb3+ ions, bright cyan-to-green color-tunable emissions were obtained in CLZA:1%Ce3+,xTb3+ phosphors with an increase of the Tb3+ doping concentration, and the corresponding CIE color coordinates changed from (0.156, 0.249) to (0.341, 0.533) when the Tb3+ doping concentration increased from x = 0 to 90%. The internal quantum efficiency reached up to 69%. Moreover, temperature-dependent spectra demonstrated good thermal stability and color stability of the as-prepared Ce3+/Tb3+ doped Ca2LuZr2(AlO4)3 phosphors. In addition, a white LED device was fabricated by coating the blend of the as-prepared cyan-emitting CLZA:1%Ce3+,10%Tb3+ phosphor and commercial red phosphor CaAlSiN3:Eu2+ on a 400 nm LED chip, which exhibited bright warm-white light with CIE coordinates of (0.425, 0.376), low correlated color temperature of 2,975 K and high color rendering index value of Ra = 83 and R9 = 89. These results revealed that the CLZA:Ce3+,Tb3+ phosphors could be excellent candidates for the application of near-UV-excited warm-white LEDs.