Because of their extraordinary physical properties, low-dimensional materials including graphene and gallium selenide (GaSe) are promising for future electronic and optoelectronic applications, particularly in transparent-flexible photodetectors. Currently, the photodetectors working at the near-infrared spectral range are highly indispensable in optical communications. However, the current photodetector architectures are typically complex, and it is normally difficult to control the architecture parameters. Here, we report graphene-GaSe heterojunction-based field-effect transistors with broadband photodetection from 730-1550 nm. Chemical-vapor-deposited graphene was employed as transparent gate and contact electrodes with tunable resistance, which enables effective photocurrent generation in the heterojunctions. The photoresponsivity was shown from 10 to 0.05 mA/W in the near-infrared region under the gate control. To understand behavior of the transistor, we analyzed the results via simulation performed using a model for the gate-tunable graphene-semiconductor heterojunction where possible Fermi level pinning effect is considered.