I don't believe that the spikes are actually passing through the filter, they are rather bypassing it due to unsuitable circuit layout, or are simply measurement artefacts caused by the probe ground wire.
In case that the spikes go actually through the filter, which can happen if the OP has low bandwidth and respective high output impedance a higher frequency, extending the circuit with an additional passive low pass to third order would help.
Remember that I suggested a sallen key filter with gain, but don't know if your filter design software supports it.