The operating principle of Pirani pressure sensors is based on the pressure dependence of a suspended strip's electrical conductivity, caused by the thermal conductance of the surrounding gas which changes the Joule heating of the strip. To realize such sensors, not only materials with high temperature dependent electrical conductivity are required, but also minimization of the suspended strip dimensions is essential to maximize the responsivity and minimize the power consumption. Due to this, nanomaterials are especially attractive for this application. Here, we demonstrate the use of a multi-layer suspended graphene strip as a Pirani pressure sensor and compare its behavior with existing models. A clear pressure dependence of the strip's electrical resistance is observed, with a maximum relative change of 2.75% between 1 and 1000 mbar and a power consumption of 8.5 mW. The use of graphene enables miniaturization of the device footprint by 100 times compared to state-of-the-art. Moreover, miniaturization allows for lower power consumption and/or higher responsivity and the sensor's nanogap enables operation near atmospheric pressure that can be used in applications such as barometers for altitude measurement. Furthermore, we demonstrate that the sensor response depends on the type of gas molecules, which opens up the way to selective gas sensing applications. Finally, the graphene synthesis technology is compatible with wafer-scale fabrication, potentially enabling future chiplevel integration with readout electronics.