Finland's Valtteri Bottas will remain as the former champions' other driver.
"When I was a kid I always dreamed about racing for Ferrari, Williams or McLaren," Massa said in a statement before addressing assembled Williams employees at the factory in central England.
"I'm glad to be signing with another icon of the sport following my time at Ferrari."
The 32-year-old has raced for Ferrari since 2006 with 11 grands prix wins during that period. He missed out on the world championship by a single point in 2008 to Britain's Lewis Hamilton.
He follows in the footsteps of the late triple champion Ayrton Senna, Nelson Piquet, Rubens Barrichello and Bruno Senna as Brazilians to have driven for Williams.
"He is an exceptional talent and a real fighter on the racetrack," said team principal Frank Williams. "He also brings a wealth of experience as we begin a new chapter in our story."
Ferrari have already announced that 2007 world champion Kimi Raikkonen, who is out of contract with Lotus at the end of the season, will replace the Brazilian at the Italian team.
Maldonado brought substantial backing from Venezuelan oil company PDVSA to Williams when he arrived in 2011 but has become frustrated with the former champions, who have scored just one point this season.
Williams thanked him for his efforts, in particular his victory in last year's Spanish Grand Prix.
"This announcement is a key step towards our goal of returning Williams to the front of the grid, and part of our ongoing plans to ensure we are stronger in 2014 and beyond," said deputy principal Claire Williams.
"The stability of having both our drivers on multi-year contracts, Mercedes-Benz as our engine partner and a strong commercial base all contribute to the future success of the Williams F1 team."
Massa said 2014, with a new V6 engine being introduced and major rule changes, will be a fresh start.
"I hope my experience will be useful in helping the team in its effort to move on from a difficult period," he said.
"I'm fully prepared to work with everyone at Grove to make sure we find the right direction. I'm highly motivated to start working hard from the very beginning in what is an exciting new challenge in my career."
(Reporting by Alan Baldwin, editing by Tony Goodson)
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