We report experimental observation of room-temperature current gains as large as 50 in a novel transistor grown in the InAs/GaSb/AlSb material system. Due to the unique degree of flexibility this material system offers in choosing band alignments, the base and collector terminals are separated by a quantum barrier while electrons traveling between the emitter and collector terminals do not tunnel across any classically forbidden regions, even though a quasi-bound state exists in the quantum well collector. This asymmetry in current conduction between the terminals of the device leads to transistor action: applying a bias to the base terminal electrostatically modulates the emitter-collector current through Stark shifts of the energy levels in the quantum well collector, while the quantum barrier between the base and collector terminals suppresses the base current. Because transport through the structure is dependent on resonant transmission, this novel transistor holds promise for the fabrication of high-speed circuits.