A high-precision distributed time delay measurement in a chirped pulse phase optical time domain reflectometry (CP φ-OTDR) system based on a random fiber grating array is proposed and demonstrated, in which a temperature-induced refractive index and fiber dimension change associated time delay could be measured for distributed temperature sensing. The random fiber grating array includes many inscribed refractive index change locations at periods of sub-micron. When laser pulses are launched into the fiber grating, the backscattered light possesses many unique localized speckle patterns at different locations. These patterns change with temperature, and hence the backscattering spectral response will change accordingly. By measuring the localized speckle pattern change due to the change of the temperature over the chirped pulse spectrum and performing cross-correlation calculation, we can realize distributed temperature measurements in real time using a megahertz bandwidth distributed feedback laser. Unlike a conventional φ-OTDR sensing system which measures a distributed phase change along the fiber using an ultra-narrow linewidth laser, the distributed time delay presented in this Letter is directly measured in real time. It is shown that the time-resolved localized pattern trace is stable with very small fluctuation, thanks to the enhanced inhomogeneity and reflectivity. The minimum detectable temperature variation is about 0.028°C at meter order of magnitude spatial resolution.