The 18-year-old has agreed a five-year contract with the Bundesliga champions after completing a medical in Munich on Monday.
"Renato Sanches has been observed by FC Bayern over a long period," CEO Karl-Heinz Rummenigge said in a media release.
"We are fortunate to have him committed to our club in spite of well-known international competition."
Benfica later confirmed that the initial fee of €35m could rise to €80m, with an addition €45m depending on success-related bonuses.
Deputy chairman Jan-Christian Dreesen confirmed to Bayern's official website: "The transfer fee for the current international, who was voted the Shooting Star of Portuguese Football in his home country yesterday is €35million, plus possible bonus payments that could become due if for example the player is named in the World Team of the Year or FIFA World Player of the Year."
Sanches, who only made his senior Benfica debut in October, has emerged as one of the most highly regarded young players in Europe this season.
Manchester United have long been reported to be tracking the midfielder, with rumours on Monday suggesting a move to Old Trafford could still be possible.
Sanches, capped twice by Portugal, has made 24 Primeira Liga and six Champions League appearances for Benfica this season, scoring twice.
He can bow out as a champion with top-of-the-table Benfica hosting Nacional on the final day of the Portuguese season on Sunday.