Accepting the award Michael Sugar said: "This film gave a voice to survivors and this Oscar amplifies that voice, which we hope will become a choir that will resonate all the way to the Vatican".
He added: "Pope Francis is trying to protect the children and restore the faith."
Thought of as the top honour handed out by the Academy, Spotlight was not widely expected to win best picture and the film won only one other award on the night, taking the gong for best original screenplay.
The movie, which was nominated for six Oscars, looks at the scandal of sex abuse in the Catholic Church in Boston, and the collusion and reluctance of the local community to face up to it.
The four awards it missed out on were actor in a supporting role for Mark Ruffalo, actress in a supporting role for Rachel McAdams, film editing for Tom McArdle, and directing for Tom McCarthy.
McCarthy and Josh Singer jointly won the original screenplay award for Spotlight.
Speaking backstage, McCarthy said: "It's electric, it's really exciting. With a project like this, you don't think about awards. It's truly thrilling and rewarding."
When an electrical extension cord fell down from the ceiling above his and Singer's heads as they spoke to reporters, McCarthy quipped: "That was the Catholic Church."
Speaking at Spotlight's UK premiere last month, Ruffalo said he was surprised that it took so long for the crimes to be exposed, but said he thought Pope Francis is more willing than his predecessors to address this past abuse.
He told the Press Association: "It's so outrageous, and it clearly has been going on for a very long time.
"And so a lot of people had to turn the other way for this to have been under wraps for so long. I'm just happy that now we are having this conversation and these poor, suffering souls who were the victims of these crimes finally get a voice and finally have been noticed.
"This is not particular to Boston or to America - it has clearly been happening everywhere. And I feel like we have a Pope now who is probably more willing than any other pope in the past decades to start to address this issue."