Most compressed sensing algorithms do not account for the effect of saturation in noisy compressed measurements, though saturation is an important consequence of the limited dynamic range of existing sensors. The few algorithms that handle saturation effects either simply discard saturated measurements, or impose additional constraints to ensure consistency of the estimated signal with the saturated measurements (based on a known saturation threshold) given uniform-bounded noise. In this paper, we instead propose a new data fidelity function which is directly based on ensuring a certain form of consistency between the signal and the saturated measurements, and can be expressed as the negative logarithm of a certain carefully designed likelihood function. Our estimator works even in the case of Gaussian noise (which is unbounded) in the measurements. We prove that our data fidelity function is convex. We moreover, show that it satisfies the condition of Restricted Strong Convexity and thereby derive an upper bound on the performance of the estimator. We also show that our technique experimentally yields results superior to the state of the art under a wide variety of experimental settings, for compressive signal recovery from noisy and saturated measurements.