We explore the use of integral field spectroscopy (IFS) for observing extrasolar planet transits. Although this technique should find its full potential in space-based observations (e.g., James Webb Space Telescope, Terrestrial Planet Finder), we have tested its basics with ground-based time-series observations of HD 209458b obtained with the William Herschel Telescope optical fiber system INTEGRAL during a transit in 2004 August 17/18. For this analysis we have used 5550 spectra (from a potential of ~30,000) obtained in 150 exposures during a period of more than 7 hr. We have found that IFS offers three fundamental advantages with respect to previously used methods (based on imaging or standard slit spectroscopy). First, it improves the effective signal-to-noise ratio in photon-limited observations by distributing the light coming from the star into the two dimensions of the detector. Second, this type of IFS data allows us to ``autocalibrate'' instrumental and background effects. Third, since the star image characteristics (i.e., seeing, spatial shifts, etc.) as well as its photometric properties are extracted from the same data cube, it is possible to decorrelate photometric instabilities induced by point-spread function (or instrument) variations. These data have also allowed us to explore the accuracy limits of ground-based relative spectrophotometry. This was done using a photometric index that probes the Na D lines, for which we obtained a nominal 1 σ error of ~1.0 × 10-4. This result, based on observations of only one transit, indicates that this type of ground observation can constrain the characterization of the transmission spectrum of extrasolar planets, especially if they cover multiple transits under good weather conditions. The present observations are compatible with no extra Na D depression during the transit. Although this result seems to be inconsistent with the recently reported Hubble Space Telescope STIS findings, we point out its limited statistical meaning: the results disagree within 1 σ, but agree within 2 σ. This method requires careful and specific reductions, and details about this process are given. We also give some recommendations to instrument developers in order to enhance the efficiency of the method.