A mid-infrared sensor is proposed in which an intersubband quantum-dot (QD) detector is integrated with an avalanche photodiode (APD) through a tunnel barrier. In the proposed three-terminal device, the applied biases of the QD and the APD are controlled separately; this feature permits the control of the QD's responsivity and dark current independently of the operational gain of the APD. It is shown theoretically that the proposed device can achieve a higher signal-to-noise ratio (SNR) over the QD detector without the APD component. Indeed, prior studies have revealed that although a heterostructure barrier lowers both the dark current and the photocurrent of the QD detector, the barrier has a greater impact on the dark current. Thus the dark-current-limited SNR is enhanced in the presence of the barrier. However, due to the reduced photocurrent, the SNR may not achieve its potential in the presence of Johnson noise, which may become dominant, for example, at low integration times or when detecting ultra-weak signals. In the proposed device, the APD component provides the necessary photocurrent gain required to elevate the SNR to the dark-current limit. This improvement, however, comes at a slight penalty in the SNR, due to the excess noise introduced by the APD. In this paper, guidelines for the SNR improvement are discussed in terms of the QD's operational bias voltage and the required APD gain. The higher SNR could be used to obtain a higher sensitivity at the same temperature, or to achieve a comparable performance at higher operating temperatures.