A tunable absorber made of a nanodisk-based metasurface is proposed to realize a narrowband shortwave-infrared (SWIR) perfect absorption. By introducing a phase-change material Ge2Sb2Te5 (GST) layer, we produce a selective and active control of the optical response. It is found that the narrowband absorption of 99.9% can be achieved for amorphous GST (aGST) with a modulation depth of 54.6% at 1931 nm, which is attributed to the strong electric dipole resonance in the germanium nanodisks. Moreover, under the aGST state, the full width at half-maximum of 22 nm can be acquired for a normal TM-polarized wave, and such a nanodisk-based absorber enables a tunable operating wavelength by adjusting the geometrical parameters to realize the spectral selectivity. In addition, the nanodisk-based metasurface nanostructure, combined with a dielectric Bragg reflector with alternately stacked SiO2 and TiO2 layers, can realize the SWIR dual-band absorption for aGST and single-band absorption for crystalline GST through the adjustment of electric and magnetic resonances. The designed absorbers have the potential applications in tunable absorption filter, thermal sensing, and optical signal processing.