Miller's movie — based on the real-life murder of Olympic wrestler Dave Schultz by multi-millionaire John du Pont — has seen "The Office" star Carell hailed by critics.
The film, in which he plays the deranged, sinister du Pont, also stars Channing Tatum and Mark Ruffalo as Mark and Dave Schultz, two wrestling-champion brothers.
"It's really something to be supported, and to have people who have faith in you, and to come out the other side," Miller told the audience at the festival hall in Cannes as he accepted his prize.
Speaking afterwards, he lauded Carell's courage in taking on the role.
"I wonder if it can really be communicated how courageous that is to show up knowing that the survivors of the family and the people who lived the story are going to be around set, and that the movie is going to be scrutinised," he told reporters.
Both Mark Schultz and Dave's widow Nancy came on set to help the actors and filmmakers during the shoot.
Carell said earlier this week that his first encounter with Nancy had been very awkward as he was decked out as his character du Pont when they met.
Miller also praised Tatum and Ruffalo, adding he never wanted to make a film without the latter, who "is an amazing actor" and "an amazing person."
"Foxcatcher" opens on Mark (Tatum), who has always lived in the shadow of his loving elder sibling Dave (Ruffalo).
So when du Pont asks him out-of-the-blue to move to his sprawling estate and help put together a wrestling team for the 1988 Seoul Olympics, the insecure sportsman jumps at the chance.
Mark soon develops a father-son relationship with the erratic du Pont, who nevertheless turns on him and calls on his more confident brother to train the team instead.
Resentful at first, Mark eventually grows close to his brother again whilst du Pont becomes alienated from them, culminating in the tragic murder that made headlines in 1996 when the heir shot Dave Schultz.
The New York-born director has only made three feature films, but already he is a regular at international award ceremonies.
He directed his childhood friend Philip Seymour Hoffman to Oscar glory with "Capote", his 2005 biopic of author and playwright Truman Capote.
Hoffman also won a BAFTA and Golden Globe for his performance.
Then came "Moneyball", a 2011 biographical baseball drama which also got several award nominations.
At a press conference in Cannes , Miller choked up about Hoffman when a journalist asked about his ability to make actors disappear into roles, pointing to Carell and Hoffman's star turn as Capote.
"It makes me emotional. The last time I saw you, I was more emotional than I wanted to ever be in front of people," he said, before pausing and stumbling for words.
Miller took eight years to prepare "Foxcatcher", researching the story by meeting those who knew the two brothers, and said the film had only properly been finished two weeks before.