A novel method developed to increase the temporal resolution of x-ray streak cameras is described. The method is analogous to the time-correlated single-photon-counting technique, which is commonly used in atomic physics. By use of short-pulse x-ray radiation from a laser-produced plasma, generated by an ultrafast laser, it is shown that a standard x-ray streak camera with a nominal temporal resolution of >5ps can yield a temporal response of 1.6 ps. The readout technique also removes temporal jitter with respect to the triggering laser. Capabilities and limitations of the technology are discussed.