The supporter was taken ill during Pompey's 2-1 defeat to the Cherries and was unable to be revived after being taken to hospital.
The League One club released a statement on Saturday night, offering their condolences to the supporter's family.
It read: "Portsmouth Football Club are sad to announce that a Pompey fan has passed away after falling ill at today's match.
"The incident occurred during the half-time break and the start of the second half was delayed while he received treatment.
"Despite valiant attempts by medical staff - who were quickly on hand to administer CPR - and following transportation to QA Hospital, he was unable to be revived.
"Next of kin have been notified and our thoughts are with them at this time."