Godson leaves the paper after less than two years as business editor in charge of a team of 15.
The journalist of 17 years effectively replaces Goldman's former corporate affairs chief Lucas van Praag, who was promoted to head of global corporate comms in January this year.
Godson said his leaving date from The Sunday Times is yet to be determined, but confirmed he hoped to be in place at Goldman by the new year.
On his move into PR, Godson said: 'After years of being a voyeur, it's good to see what's happening on the inside.'
While he said his decision was not swayed by an attractive pay packet, speculation suggests the role is worth up to £250,000 a year.
At Goldman he will oversee a team of around 12 communications and marketing staff, split between offices in London and Frankfurt.
He is also set to join the bank's European management committee and will report to Van Praag.
'Goldman is a special case - it is very cohesive and coherent and all work on the same team,' said Godson.
Godson is the latest in a line of City journalists to move into PR in the last year, following former Sunday Telegraph City editor Neil Bennett's appointment as Gavin Anderson's London CEO in May, and ex-Sunday Times' City editor Kirstie Hamilton's departure to join Tulchan Communications in March.