Four hundred members of the Communication Workers Union (CWU) were planning a 24-hour walkout in a dispute over flexible working hours.
The centre processes an estimated 1.2m letters and parcels every day.
The strike was due to begin in the afternoon and managers had been drafted in to keep deliveries on track.
Phil Chadwick, of the CWU, said: "We are very close to an agreement and we trust that the Royal Mail will see our actions in a positive climate and put an end to uncertainty for the public and for our members."
Postal workers held an unofficial strike at the Exeter centre in August, disrupting deliveries to nearly 500,000 homes in the EX and TQ postcode areas of Devon.
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