Existing convex relaxation-based approaches to reconstruction in compressed sensing assume that noise in the measurements is independent of the signal of interest. We consider the case of noise being linearly correlated with the signal and introduce a simple technique for improving compressed sensing reconstruction from such measurements. The technique is based on a linear model of the correlation of additive noise with the signal. The modification of the reconstruction algorithm based on this model is very simple and has negligible additional computational cost compared to standard reconstruction algorithms, but is not known in existing literature. The proposed technique reduces reconstruction error considerably in the case of linearly correlated measurements and noise. Numerical experiments confirm the efficacy of the technique. The technique is demonstrated with application to low-rate quantization of compressed measurements, which is known to introduce correlated noise, and improvements in reconstruction error compared to ordinary Basis Pursuit De-Noising of up to approximately 7 dB are observed for 1 bit/sample quantization. Furthermore, the proposed method is compared to Binary Iterative Hard Thresholding which it is demonstrated to outperform in terms of reconstruction error for sparse signals with a number of non-zero coefficients greater than approximately 1⁄10th of the number of compressed measurements.