Due to their periodic nature, neural oscillations might represent an optimal "tool" for the processing of rhythmic stimulus input [1-3]. Indeed, the alignment of neural oscillations to a rhythmic stimulus, often termed phase entrainment, has been repeatedly demonstrated [4-7]. Phase entrainment is central to current theories of speech processing [8-10] and has been associated with successful speech comprehension [11-17]. However, typical manipulations that reduce speech intelligibility (e.g., addition of noise and time reversal [11, 12, 14, 16, 17]) could destroy critical acoustic cues for entrainment (such as "acoustic edges" [7]). Hence, the association between phase entrainment and speech intelligibility might only be "epiphenomenal"; i.e., both decline due to the same manipulation, without any causal link between the two [18]. Here, we use transcranial alternating current stimulation (tACS [19]) to manipulate the phase lag between neural oscillations and speech rhythm while measuring neural responses to intelligible and unintelligible vocoded stimuli with sparse fMRI. We found that this manipulation significantly modulates the BOLD response to intelligible speech in the superior temporal gyrus, and the strength of BOLD modulation is correlated with a phasic modulation of performance in a behavioral task. Importantly, these findings are absent for unintelligible speech and during sham stimulation; we thus demonstrate that phase entrainment has a specific, causal influence on neural responses to intelligible speech. Our results not only provide an important step toward understanding the neural foundation of human abilities at speech comprehension but also suggest new methods for enhancing speech perception that can be explored in the future.