We used the Hubble Space Telescope (HST) Wide Field Camera 3 (WFC3) near-infrared camera to image the host galaxies of a sample of 11 luminous, dust-reddened quasars at z similar to 2-the peak epoch of black hole growth and star formation in the universe-to test the merger-driven picture for the coevolution of galaxies and their nuclear black holes. The red quasars come from the FIRST+2MASS red quasar survey and a newer, deeper, UKIDSS +FIRST sample. These dust-reddened quasars are the most intrinsically luminous quasars in the universe at all redshifts, and they may represent the dust-clearing transitional phase in the merger-driven black hole growth scenario. Probing the host galaxies in rest-frame visible light, the HST images reveal that 8/10 of these quasars have actively merging hosts, whereas one source is reddened by an intervening lower-redshift galaxy along the line of sight. We study the morphological properties of the quasar hosts using parametric Sersic fits, as well as nonparametric estimators (Gini coefficient, M-20, and asymmetry). Their properties are heterogeneous but broadly consistent with the most extreme morphologies of local merging systems such as ultraluminous infrared galaxies. The red quasars have a luminosity range of log(Lbol)= 47.8-48.3 (erg s(-1)), and the merger fraction of their hosts is consistent with merger-driven models of luminous active galactic nuclei activity at z = 2, which supports the picture in which luminous quasars and galaxies coevolve through major mergers that trigger both star formation and black hole growth.