Fox-Pitt was held up by a security strike at Copenhagen Airport as he returned from the European Championships in Malmo where he had claimed individual bronze aboard Chilli Morning.
And the three-time Olympic medallist opened his campaign at Burghley by placing ninth with Neuf des Coeurs after scoring 49.0 in the first of two days of dressage.
Fox-Pitt, who won team silver at the London 2012 Olympics last summer, will return aboard Parklane Hawk on day two with Cohen currently two places ahead of him in seventh.
Cohen, riding Treason, scored 48.0 for her dressage while Piggy French ensured there were three Brits in the top ten after the first day with a score of 49.2 aboard Westwood Mariner.
Ingrid Klimke, who won silver at the Europeans last week, currently leads after scoring 39.0 with Jonathan Paget of New Zealand and fellow German Andreas Dibowski both second with 42.0.
Of those to compete on the second day of dressage are Fox-Pitt's fellow Olympic silver medallists from London 2012 Nicola Wilson and Kristina Cook.