Eddie had seen off competition from leading men whose roles were equally transformative – Steve Carell in Foxcatcher, Jake Gyllenhall for Nightcrawler and fellow Brits David Oyelowo in Selma, and The Imitation Game's Benedict Cumberbatch.
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Eddie Redmayne was named Best Actor at the Golden Globes 2015
"I've had to spend most of the evening stopping myself from falling over and letching over actors I've long admired," the overwhelmed star told the audience in his speech, before personally thanking the Hawking family.
"This was a huge privilege," Eddie said. "Stephen, Jane, the Hawking family allowed us into their lives and to tell their story." He also praised his co-star Felicity Jones, describing her as "the most formidable actress" who had made him raise his game.
The Theory of Everything star apologised to new wife Hannah Bagshawe in his speech
Eddie took the opportunity to apologise to his new wife Hannah Bagshawe at the end of his speech. "We had to cut our honeymoon short to come to Los Angeles," he said of his bride, who proudly watched from her seat in the auditorium. "I thank the HFPA for giving me and us a night at the end of our honeymoon we will all remember."
Also enjoying success on the night was Julianne Moore, who received the Best Actress award for her role in Still Alice.
Best Actress Julianne Moore
Julianne beat Reese Witherspoon in Wild, Jennifer Aniston in Cake and Gone Girl's Rosamund Pike to take home the trophy for the drama, in which she plays a linguistics professor diagnosed with early onset Alzheimer's disease.
Boyhood, meanwhile, swept the board at the 72nd Golden Globes. The innovative coming-of-age movie, which was took 12 years to make, won Best Film with Richard Linklater named Best Director and Patricia Arquette Best Supporting Actress.
Downton Abbey's Joanna Froggatt (right) celebrates with co-star Laura Carmichael
Other winners on the night included British actresses Joanne Froggatt – named Best Supporting Actress in a TV series for Downton Abbey – and Ruth Wilson, who collected the award for Best Actress in a US TV drama for The Affair.
Comedy drama Birdman picked up two of the seven awards it had been nominated for, including one for Michael Keaton as best actor in a comedy or musical.
The Grand Budapest Hotel won best comedy or musical film, with Amy Adams named best actress in a comedy or musical for her role in Big Eyes.
Despite being nominated for five awards, Alan Turing biopic The Imitation Game came away empty-handed, as did Gone Girl, which had received four nods.