Hodgson resigned from the post in the immediate aftermath of the Three Lions' shock exit to Iceland at the last-16 stage of Euro 2016.
Allardyce was the first candidate to be interviewed for the vacancy, having also been considered in 2006 before the FA settled on Steve McClaren to succeed Sven-Goran Eriksson after the World Cup in Germany.
Steve Bruce, who quit his position at Hull earlier in the day, also met with FA chiefs over the national-team job but is now expected to return to the Stadium of Light as Allardyce's replacement.
The 61-year-old leaves the Black Cats with immediate effect to step into his new role, which sees him take charge for the 2018 World Cup qualifiers initially.
"I am extremely honoured to be appointed England manager especially as it is no secret that this is the role I have always wanted. For me, it is absolutely the best job in English football," Allardyce told the FA's official website.
"I will do everything I can to help England do well and give our nation the success our fans deserve. Above all, we have to make the people and the whole country proud.
"While my main focus will be on the senior team and getting positive results, I want to add my influence to the great work being done across the development teams at St George's Park - a facility I have used with my previous clubs.
"I know we have talented, committed players and it is time for us to deliver."
FA chief executive Martin Glenn added: "Sam Allardyce is the right man for the England job. His excellent managerial credentials, including his ability to realise the potential of players and teams, develop a strong team ethos and embrace modern methods that enhance performance, made him the outstanding choice.
"That was underlined when we sat down to talk and we could not help but be energised by his personal perspective on England's future and how it complemented the extensive work that we are looking to build on at St. George's Park.
"Dan Ashworth, David Gill and I have carried out a thorough process in the last three weeks and ultimately we could not look beyond Sam as the ideal candidate."