We demonstrate that a spintronic terahertz (THz) emitter can be driven by a chirped-pulse beating scheme to generate narrowband THz pulses, with continuous tuning of the frequency and linewidth by simply adjusting the laser chirp and/or the time delay between chirped pulses. As supported by model calculations, temporal shaping of the drive laser pulses can be exploited to manipulate the ultrafast demagnetization dynamics in the thin-film emitter, modulating the spin-polarized current in the ferromagnetic layer to access multi-cycle THz emission. Using a regenerative amplifier laser system with 50 fs transform-limited pulses chirped to 6 ps, we demonstrate narrowband THz generation over a frequency range from 0.4 to 2.3 THz, in addition to linewidths down to 40 GHz using 12 ps chirped pulses. Our proof-of-concept results pave the way to future narrowband THz sources with subgigahertz linewidth and center frequencies continuously tunable from 0.1 to 30 THz. By combining with the advantageous properties of spintronic THz emitters, from straightforward implementation to flexible polarization control, these sources open up opportunities for narrowband applications over the entire THz spectral range.