In this paper, we propose a robust compressive sampling approach for wideband spectrum sensing in cognitive radios in the presence of non-Gaussian noise. Wideband cognitive radios can be subjected to heterogeneous spectral activities from various sources rendering the resultant noise non- Gaussian. While conventional detectors (e.g. least-squares estimates) are known to be sensitive to the Non-Gaussian nature of noise, the proposed compressive sampling based robust detector is shown to overcome that limitation leading to better signal activity detection under such conditions. The proposed robust detector combines the Huber cost (loss) function with an l₁-norm constraint for wideband spectrum sensing with a smaller number of samples (compared to Nyquist rate sampling). Note that, while the Huber cost function robustify our approach against non- Gaussian noise, the l₁-norm regularization term ensures the sparsity in the signal reconstruction. It is shown that our proposed robust method outperforms the conventional Periodogram when applied to noisy signals with a smaller number of samples.