An approach to generating dual-chirp microwave waveforms (DCMWs) is proposed and experimentally demonstrated. The proposed scheme consists of a typical semiconductor slave laser (SL), which is subject to a dual-beam optical injection from two master lasers with one being positively detuned (${\rm ML}_1$ML1) and the other negatively detuned (${\rm ML}_2$ML2) from the SL. Under proper injection conditions, the SL operates in the so-called Scenario B of dual-beam injection. After optical-to-electrical conversion, a dual-frequency microwave signal can be generated with one of its two frequencies increasing linearly and the other decreasing linearly as the ${\rm ML}_1$ML1 injection strength is increased. By incorporating a fast injection strength controller (formed by an intensity modulator and an electrical control signal), a DCMW with a large time-bandwidth product can be generated. In the experimental demonstration, a DCMW with a temporal period of 1 µs has been obtained. This simultaneously offers an up-chirp (13.4-20.2 GHz) and a down-chirp (27.3-20.5 GHz), and its frequency tunability has been achieved by simply adjusting the injection parameters. Furthermore, the auto-ambiguity function of the generated DCMW has also been investigated, which proves that the proposed scheme has the ability to improve the range-velocity resolution and, thus, could be promising for use in radar systems.