Electrical and noise properties of graphene contacts to AlGaN/GaN heterostructures were studied experimentally. It was found that graphene on AlGaN forms a high-quality Schottky barrier with the barrier height dependent on the bias. The apparent barrier heights for this kind of Schottky diode were found to be relatively high, varying within the range of φb = (1.0–1.26) eV. AlGaN/GaN fin-shaped field-effect transistors (finFETs) with a graphene gate were fabricated and studied. These devices demonstrated ~8 order of magnitude on/off ratio, subthreshold slope of ~1.3, and low subthreshold current in the sub-picoamperes range. The effective trap density responsible for the 1/f low-frequency noise was found within the range of (1–5) · 1019 eV−1 cm−3. These values are of the same order of magnitude as reported earlier and in AlGaN/GaN transistors with Ni/Au Schottky gate studied as a reference in the current study. A good quality of graphene/AlGaN Schottky barrier diodes and AlGaN/GaN transistors opens the way for transparent GaN-based electronics and GaN-based devices exploring vertical electron transport in graphene.