Palmer struck twice deep into time added on to save Chelsea who were on the verge of a 3-2 home defeat.
He completed his hat trick in the 11th minute of stoppage time with what was the latest winning goal in Premier League history.
Goals by Alejandro Garnacho and Bruno Fernandes before half-time and another by Garnacho in the 67th minute turned the tide towards the visitors.
But Palmer stroked home his second spot kick and then blasted a deflected shot into the net in the dying seconds to send Stamford Bridge into raptures.
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