Radio frequency (RF) spike noise is a common source of exogenous image corruption in MRI. Spikes occur as point-like disturbances of $k$-space that lead to global sinusoidal intensity errors in the image domain. Depending on the amplitude of the disturbances and their locations in $k$-space, the effect of a spike can be significant, often ruining the reconstructed images. Here we present both a spike detection method and a related data correction method for automatic correction of RF spike noise. To detect spikes, we found the $k$-space points that have the most significant effect on the total variation of the image. To replace the spikes, we used a compressed sensing reconstruction in which only the points thought to be corrupted are unconstrained. We demonstrated our technique in two cases: (1) in vivo gradient echo brain data with artificially corrupted points and (2) actual, complex scanner data from a whole-body fat-water imaging gradient echo protocol corrupted by spikes at uncertain locations. Our method allowed near-perfect detection and correction with no human intervention. We calculated Matthews correlation coefficients and sensitivities above 0.95 for a maximum of 0.78\% corruption in synthetically corrupted in vivo brain data. We also found specificities above 0.9994.