Finch has headed up the coach firm National Express since 2010.
Before than he was group CEO of Tube Lines which owns and maintains three of the London Underground lines, and was group finance director and chief operating officer at First Group.
He is expected to join the housebuilder at the end of the year, on an annual salary of £725,000.
Finch succeeds David Jenkinson who announced his plans to leave in February.
Persimmon chairman Roger Devlin said: “Dean is a seasoned, well-respected and proven chief executive with an exceptional record.
“In his current role he has delivered substantial strategic and operational progress over a sustained period, delivering value for all stakeholders in the business while developing a distinct and cohesive culture, focused on customer care and service.
“The board believes that Dean is a great fit for Persimmon and is well qualified to lead the business into the next phase as we continue to drive a programme of change to become the leading volume builder of good value, quality family homes throughout the UK.”
Devlin also thanked Jenkinson for his contribution to Persimmon over many years, and “particularly for his decisive leadership during the Covid-19 outbreak”.
“Dave has committed to lead the business until the conclusion of the current financial year in December.”