The Carina Nebula harbors a large population of high-mass stars, including at least 75 O-type and Wolf-Rayet (WR) stars, but the current census is not complete since further high-mass stars may be hidden in or behind the dense dark clouds that pervade the association. With the aim of identifying optically obscured O- and early B-type stars in the Carina Nebula, we performed the first infrared spectroscopic study of stars in the optically obscured stellar cluster Tr 16-SE, located behind a dark dust lane south of {eta} Car. We used the integral-field spectrograph KMOS at the ESO VLT to obtain H- and K-band spectra with a resolution of R~4000 ({Delta}{lambda}~5{AA}) for 45 out of the 47 possible OB candidate stars in Tr 16-SE, and we derived spectral types for these stars. We find 15 stars in Tr 16-SE with spectral types between O5 and B2 (i.e. high-mass stars with M>=8M_{sun}_), only two of which were known before. An additional nine stars are classified as (Ae)Be stars, i.e. intermediate-mass pre-main sequence stars, and most of the remaining targets show clear signatures of being late-type stars and are thus most likely foreground stars or background giants unrelated to the Carina Nebula. Our estimates of the stellar luminosities suggest that nine of the 15 O- and early B-type stars are members of Tr 16-SE, whereas the other six seem to be background objects. Our study increases the number of spectroscopically identified high-mass stars (M>=8M_{sun}_) in Tr 16-SE from two to nine and shows that Tr 16-SE is one of the larger clusters in the Carina Nebula. Our identification of three new stars with spectral types between O5 and O7 and four new stars with spectral types O9 to B1 significantly increases the number of spectroscopically identified O-type stars in the Carina Nebula.