"La La Land", which also won Chazelle a best director Bafta, is expected to sustain its momentum at the Oscars on February 26 where it has scored a record 14 nominations.
Veteran director Ken Loach's "I, Daniel Blake" was named outstanding British film at the ceremony on Sunday night.
Casey Affleck was named best actor for "Manchester by the Sea" while its director Kenneth Lonergan won the award for best original screenplay.
While the Golden Globes were dominated by Meryl Streep's moving speech criticising US President Donald Trump, he was not mentioned that much at the Baftas though Stone referred to him indirectly while accepting her best actress trophy.
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"This country - and the US, and the world - seems to be going through a bit of a time, just a bit.
"In a time that's so divisive, I think it's so special we were able to come together tonight thanks to Bafta, to celebrate the positive gift of creativity and how it can transcend borders and how it can help people to feel a little less alone," Stone said.
The actor seemed stunned, saying, "Wow, that just happened."
Patel said the film is about a love that "transcends borders, race, colour, anything".
"Arrival", which was nominated for nine Baftas alongside "Nocturnal Animals", won just one trophy for best sound while the Tom Ford-directed movie had to return empty handed.
Loach, director of "I, Daniel Blake" which deals with the
He apologised for making a political speech so early on, but said: "Thank you to the academy for endorsing the truths of what the film says, which hundreds and thousands of people in this country know.
"The most vulnerable and poorest are treated by the government with a callous brutality that is disgraceful, a brutality that extends to keeping out refugee children we promised to help and that's a disgrace too."
After Golden Globes, Davis once again paid tribute to her late father, who worked as a janitor and horse groomer, in her speech.
"August (Wilson) answers that question so brilliantly, because what he did is he said that our lives mattered as African-Americans."
'Spider Man' star Tom Holland won the EE Rising Star award, the only trophy at Baftas that is voted for by public.