The FA added that all FA Cup rounds will be played on weekends from next term, with the fourth and fifth rounds, plus quarter-finals, to be held exclusive of Premier League matches.
"From the first round proper, the competition will be played without replays," the FA said.
"The current format, which has no replays from the fifth round onwards, has been extended throughout the 'Competition Proper' in light of changes to the calendar driven by the expanded UEFA competitions.
"The FA Cup final will now take place on the penultimate weekend of the Premier League season, on an exclusive Saturday with no Premier League fixtures taking place on the same day."
The mid-season break was also removed from the calendar, in order to allow the Premier League to start in mid-August and provide a longer close-season break.
The gradual removal of replays from the later FA Cup rounds was criticised, with some commentators saying it is another erosion of the importance of the world's oldest domestic club competition.
In the past, smaller clubs drawn at home to top-tier opposition in the FA Cup have enjoyed financial windfalls from securing replays.
"This new agreement between the FA and the Premier League strengthens the FA Cup and gives this very special tournament exclusive weekends in an increasingly busy calendar," Mark Bullingham, CEO of England's FA, said in a statement.
"The new schedule ensures the magic of the cup is protected and enhanced, while working for the whole of the English game."
As part of the agreement, the Premier League will increase the funding it provides to the English football pyramid, giving an additional 33 million pounds ($41.15 million) towards developing the grassroots game.
The Premier League already provides 100 million pounds in funding to teams in lower leagues.
"This (investment) will improve facilities for communities and lower league clubs across the country, through the Football Foundation and Premier League Stadium Fund," Premier League Chief Executive Richard Masters said.
This season's FA Cup semi-finals will take place on the weekend, with holders Manchester City facing Chelsea on Saturday and Coventry City taking on Manchester United on Sunday.
