In the Third-Generation Partnership Project (3GPP) Long-Term Evolution (LTE) specifications, several 2-D positioning methods have been proposed to support location-based services (LBSS). It has been shown that the observed time difference of arrival (OTDOA) method provides the best result. With the advent of the fifth-generation cellular systems, 3-D beamforming will be introduced, and 2-D positioning may not be sufficient for future applications. Although the conventional OTDOA method can be used in 3-D positioning, it requires more base stations (BSS) in cooperation and often performs poorly in typical cellular network geometry. In this paper, we propose a novel OTDOA-based method to solve the problems. First, we exploit the sparsity of the wireless channel and propose to use a compressive-sampling-based channel estimation method, which greatly enhances the precision of time-of-arrival (TOA) estimation. Then, we propose a transmit beamforming scheme such that the elevation angle of departure (AOD) can be accurately estimated. Finally, we combine the estimated TOAs and the AOD in the OTDOA scheme to conduct 3-D positioning. With the proposed scheme, the number of BSS required for 3-D positioning remains the same as that of 2-D, and the positioning accuracy greatly outperforms that of the conventional OTDOA method. Simulations show that 90% of the position error can be less than 7 m.