High-resolution Doppler spectroscopy is a powerful tool for identifying molecular species in the atmospheres of both transiting and non-transiting exoplanets. Currently, such data is analysed using cross-correlation techniques to detect the Doppler shifting signal from the orbiting planet. In this paper we demonstrate that, compared to cross-correlation methods currently used, the technique of Doppler tomography has improved sensitivity in detecting the subtle signatures expected from exoplanet atmospheres. This is partly due to the use of a regularizing statistic, which acts to suppress noise, coupled to the fact that all the data is fit simultaneously. In addition, we show that the technique can also effectively suppress contanimating spectral features that may arise due to overlapping lines, repeating line patterns, or the use of incorrect linelists. These issues can confuse conventional cross-correlation approaches, primarily due to aliasing issues inherent in such techniques, whereas Doppler tomography is less susceptible to such effects. In particular, Doppler tomography shows exceptional promise for simultaneously detecting multiple line species (e.g. isotopologues), even when there are high contrasts between such species -- and far outperforms current CCF analyses in this respect. Finally, we demonstrate that Doppler tomography is capable of recovering molecular signals from exoplanets using real data, by confirming the strong detection of CO in the atmosphere of Tau Boo b. We recover a signal with a planetary radial velocity semi-amplitude Kp = 109.6 +/- 2.2 km/s, in excellent agreement with the previously reported value of 110.0 +/- 3.2 km/s.
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