We place limits on scalar and vector unparticle couplings to electrons and neutrinos using data from reactor neutrino experiments originally designed to search for neutrino magnetic moment. Upper bounds on Standard Model lepton-scalar unparticle and lepton-vector unparticle couplings \(\lambda_0\), and \(\lambda_1\) are given for various values of the unparticle mass dimension d, and the unparticle energy scale \(\Lambda_{\cal U}\). Especially for smaller values of the mass dimension d (1 <d <1.3) these bounds are very significant and comparable to those obtained from production rates at high energy colliders. These bounds are also similar to those obtained by the analysis of the absence of the decay of the low-energy solar and reactor neutrinos.