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Asteroseismic time-series data have imprints of stellar oscillation modes, whose detection and characterization through time-series analysis allows us to probe stellar interior physics. Such analyses usually occur in the Fourier domain by computing the Lomb-Scargle (LS) periodogram, an estimator of the power spectrum underlying unevenly-sampled time-series data. However, the LS periodogram suffers from the statistical problems of (1) inconsistency (or noise) and (2) bias due to high spectral leakage. Here, we develop a multitaper power spectrum estimator using the Non-Uniform Fast Fourier Transform (mtNUFFT) to tackle the inconsistency and bias problems of the LS periodogram. Using a simulated light curve, we show that the mtNUFFT power spectrum estimate of solar-like oscillations has lower variance and bias than the LS estimate. We also apply our method to the Kepler-91 red giant, and combine it with PBjam peakbagging to obtain mode parameters and a derived age estimate of $3.97 \pm 0.52$ Gyr. PBjam allows the improvement of age precision relative to the $4.27 \pm 0.75$ Gyr APOKASC-2 (uncorrected) estimate, whereas partnering mtNUFFT with PBjam speeds up peakbagging thrice as much as LS. This increase in efficiency has promising implications for Galactic archaeology, in addition to stellar structure and evolution studies. Our new method generally applies to time-domain astronomy and is implemented in the public Python package tapify, available at https://github.com/aaryapatil/tapify.