Confined electrons collectively oscillate in response to light, resulting in a plasmon resonance whose frequency is determined by the electron density and the size and shape of the confinement structure. Plasmons in metallic particles typically occur in the classical regime where the characteristic quantum level spacing is negligibly small compared to the plasma frequency. In doped semiconductor quantum wells, quantum plasmon excitations can be observed, where the quantization energy exceeds the plasma frequency. Such intersubband plasmons occur in the mid- and far-infrared ranges and exhibit a variety of dynamic many-body effects. Here, we report the observation of intersubband plasmons in carbon nanotubes, where both the quantization and plasma frequencies are larger than those of typical quantum wells by three orders of magnitude. As a result, we observed a pronounced absorption peak in the near-infrared. Specifically, we observed the near-infrared plasmon peak in gated films of aligned single-wall carbon nanotubes only for probe light polarized perpendicular to the nanotube axis and only when carriers are present either in the conduction or valence band. Both the intensity and frequency of the peak were found to increase with the carrier density, consistent with the plasmonic nature of the resonance. Our observation of gate-controlled quantum plasmons in aligned carbon nanotubes will not only pave the way for the development of carbon-based near-infrared optoelectronic devices but also allow us to study the collective dynamic response of interacting electrons in one dimension.