The British star scooped the prize after impressing the Academy with his incredible portrayal of Stephen Hawking in The Theory of Everything.
Eddie, 33, hugged last year's Best Actress winner Cate Blanchett tightly as she presented him with the award before going on to dish out a handful of thank yous in a passionate acceptance speech.
The actor told the crowd at Hollywood's Dolby Theatre: "Thank you, thank you to the academy. I don't think I'm capable of articulating how I feel right now, but I'm fully aware I'm a lucky, lucky man."
He dedicated the award to "all of those people around the world'' battling motor neurone disease - the illness that left Professor Hawking in a wheelchair.
Brought close to tears just by saying the word 'Oscar' he continued: "It belongs to one exceptional family, Stephen, Jane, Johnathan and the Hawking children. I will be the custodian and I will promise to look after him and answer his beck and call and wait on him hand and foot.
"I wouldn't be here without my staggering partner in crime Felicity Jones and my ferocious director James March."
He also praised his "genius team" before going on to thank his family and wife Hannah.
"I also just want to thank my family and my wife Hannah - I love you so much. We've got a new fella coming to share our apartment!"
The actor beat Michael Keaton, Bradley Cooper, Steve Carell and Benedict Cumberbatch to land the top acting gong.
He becomes the eigth-youngest star to pick up the award in the Oscars' 87 year-long history. abd71
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