Star Hugh Bonneville thanked the drama’s creator, Julian Fellowes, for “giving us wonderful lines to say”.
BBC drama Doctor Foster collected two awards – best new drama and best drama performance for its star Suranne Jones.
“If anyone wants to send me tips on how to give birth that would be useful,” she said as she collected the award, thanking the drama’s writer, Mike Bartlett, for creating “a complex” character.
BBC soap EastEnders was also a double award winner. It beat rivals Coronation Street, Emmerdale and Hollyoaks to receive the prize for best serial drama.
Danny Dyer also received the best serial drama performance prize for a second consecutive year for his role as Mick Carter in the programme.
“I’m so honoured to be part of such a sublime and depressing programme that is EastEnders,” he said while collecting his award.
Anthony McPartlin and Declan Donnelly were named best TV presenter for the 15th consecutive year.
“It’s getting a bit mental now,” McPartlin said, “but we thank you from the bottom of our hearts.”
Donnelly added: “People ask us does it get boring or old – of course it doesn’t. If anything our gratitude has grown over the years as you never know when you’re going to get it again.”
Ant and Dec show I’m a Celebrity… Get Me Out of Here! won best entertainment – an award it has won every year since 2011.
BBC One show The Great British Bake Off was also named best challenge show for a second consecutive year.
Collecting the award, judge Paul Hollywood said the last series was “the best year we’ve ever had – the bakers were the best”.
Peter Kay’s Car Share beat Benidorm, Birds Of A Feather and Not Going Out to win best comedy.
Kay dedicated the award to Scottish comic Billy Connolly, “my comedy hero”.
Connolly was presented with the special recognition award at the ceremony in honour of his 50-year career.
Introduced by his friend and Oscar-winning actor Dustin Hoffman, the star paid tribute to the comic, describing him as “an artist in the truest sense of the word”.
He said: “Audiences return and new generations come to him not because he makes them laugh, but because he has the gift of inviting everyone uncritically, unconditionally, into his world.
“He exposes truth – no one looks at the world like he does; he turns the ordinary to the extraordinary and he elevates the mundane to the magnificent in the bravest of fashions.”
Receiving two standing ovations from the audience, the star joked as he walked on stage: “I’d like to thank the Catholic Church for the rhythm method of birth control – without which I wouldn’t be here at all.”
He joked with Kay for dedicating his best comedy prize to him, but then not giving the physical award away to him – prompting Kay to climb the stage to hand it over.
Other winners included X Factor winner-turned actor Shayne Ward, who was voted best newcomer for his role as Aidan Connor in Coronation Street.
Strictly Come Dancing won best talent show, with This Morning named best live magazine.
The best factual entertainment award went to Gogglebox, quiz show The Chase won best daytime and US sitcom The Big Bang Theory won best international programme.
A new award was also created this year – the impact award for best TV moment.
It went to Aidan Turner for his torso-revealing turn as Ross Poldark in the Cornish-set BBC drama series Poldark.