He was due to open in the Arthur Miller play Death of a Salesman at the Royal & Derngate in Northampton on 10 April, but died three days beforehand.
The theatre said the role of Willy Loman would now be taken by TV and film actor Nicholas Woodeson.
Shows in Northampton have been postponed and the play will now open in Cambridge in May, ahead of a UK tour.
Pigott-Smith died at the age of 70, a month after collecting an OBE for services to drama.
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He had been rehearsing the role of Willy Loman in Miller's play which was expected to open at the Northampton theatre ahead of a tour including Cambridge, Bath, Exeter, Edinburgh and Oxford.
A spokesman for the Royal & Derngate said the decision to proceed with the production and tour of Death Of A Salesman was made after consulting the "acting company, creative team and touring partners".
Pigott-Smith's replacement, Woodeson, is well-known for roles in a number of dramas including Taboo, The Living and the Dead, Rome, Mapp and Lucia, and also for film roles including in the Bond movie Skyfall.
The play will now open at Cambridge Arts Theatre on 4 May.
Ticket-holders for the postponed run in Northampton would be offered a refund, or tickets for the rescheduled run at the Royal & Derngate at the end of the tour, organisers said.
Theatre directors have confirmed a show will go on despite the death of its star, the actor Tim Pigott-Smith.