We introduce a new color selection technique to identify high-redshift, massive galaxies that are systematically missed by Lyman-break selection. The new selection is based on the H-160 (H) and Infrared Array Camera (IRAC) 4.5 mu m bands, specifically H - [4.5] \textgreater 2.25 mag. These galaxies, called “HIEROs,” include two major populations that can be separated with an additional J - H color. The populations are massive and dusty starforming galaxies at z \textgreater 3 (JH - blue) and extremely dusty galaxies at z less than or similar to 3 (JH - red). The 350 arcmin(2) of the GOODS-North and GOODS-South fields with the deepest Hubble Space Telescope (HST)/Wide Field Camera 3 (WFC3) near-infrared and IRAC data contain as many as 285 HIEROs down to [4.5] \textless 24 mag. Inclusion of the most extreme HIEROs, not even detected in the H band, makes this selection particularly complete for the identification of massive high-redshift galaxies. We focus here primarily on JH - blue (z \textgreater 3) HIEROs, which have a median photometric redshift \textless z \textgreater similar to 4.4 and stellar mass M-* 10(10.6) M-circle dot and are much fainter in the rest-frame UV than similarly massive Lyman-break galaxies (LBGs). Their star formation rates (SFRs), derived from their stacked infrared spectral energy distributions (SEDs), reach similar to 240 M-circle dot yr(-1), leading to a specific SFR, sSFR equivalent to SFR/M-* similar to 4.2 Gyr(-1), suggesting that the sSFRs for massive galaxies continue to grow at z \textgreater 2 but at a lower growth rate than from z = 0 to z = 2. With a median half-light radius of 2 kpc, including similar to 20% as compact as quiescent (QS) galaxies at similar redshifts, JH - blue HIEROs represent perfect star-forming progenitors of the most massive (M-* greater than or similar to 10(11.2) M-circle dot) compact QS galaxies at z similar to 3 and have the right number density. HIEROs make up similar to 60% of all galaxies with M-* \textgreater 10(10.5) M-circle dot identified at z \textgreater 3 from their photometric redshifts. This is five times more than LBGs with nearly no overlap between the two populations. While HIEROs make up 15%-25% of the total SFR density at z similar to 4-5, they completely dominate the SFR density taking place in M-* 10(10.5) M-circle dot galaxies, and HIEROs are therefore crucial to understanding the very early phase of massive galaxy formation.