Asteroseismic time-series data have imprints of stellar oscillation modes, whose detection and characterization through time-series analysis allows us to probe stellar interiors physics. Such analyses usually occur in the Fourier domain by computing the Lomb-Scargle (LS) periodogram, an estimator of the \textit{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. In addition, it is designed to detect strictly periodic signals but is unsuitable for non-sinusoidal periodic or quasi-periodic signals. Here, we develop a multitaper spectral estimation method that tackles the inconsistency and bias problems of the LS periodogram. We combine this multitaper method with the Non-Uniform Fast Fourier Transform (\texttt{mtNUFFT}) to more precisely estimate the frequencies of asteroseismic signals that are non-sinusoidal periodic (e.g., exoplanet transits) or quasi-periodic (e.g., pressure modes). We illustrate this using a simulated and the Kepler-91 red giant light curve. Particularly, we detect the Kepler-91b exoplanet and precisely estimate its period, $6.246 \pm 0.002$ days, in the frequency domain using the multitaper F-test alone. We also integrate \texttt{mtNUFFT} into the \texttt{PBjam} package to obtain a Kepler-91 age estimate of $3.96 \pm 0.48$ Gyr. This $36$\% improvement in age precision relative to the $4.27 \pm 0.75$ Gyr APOKASC-2 (uncorrected) estimate illustrates that \texttt{mtNUFFT} has promising implications for Galactic archaeology, in addition to stellar interiors and exoplanet studies. Our frequency analysis method generally applies to time-domain astronomy and is implemented in the public Python package \texttt{tapify}, available at \url{https://github.com/aaryapatil/tapify}.