The actress, who is set to star in the second series of the show, burst into tears on stage after being named leading actress, saying: “Oh sorry, not cool,” then hailed the show’s writer Chris Chibnall “a f***ing genius”.
Her co-star David Bradley was named best supporting actor for his role in the show which also picked up the Bafta for best drama.
Accepting his award, Bradley revealed his agent had sent him a full-size chocolate Bafta at Christmas and joked: “Now finally I can take it down from the mantelpiece and eat it.”
Speaking backstage, Bradley said the show “just became a kind of phenomenon” and praised Chibnall for “keeping all the plates twirling in the air”.
It was a good night for Geordie duo Ant and Dec who each got an award to take home when they won best entertainment programme and entertainment performance for Saturday Night Takeaway.
Accepting the second award, Declan Donnelly joked: “I’m glad we came now.”
Other big winners included Julie Walters who was given the Bafta Fellowship and told the audience her mother had said becoming an actress would see her end up in the gutter.
She said: “She was right, but what a gutter and I shared that gutter with some of the most amazing and talented people.”
Sean Harris won the leading actor award for his performance in Channel 4’s Southcliffe.
The awards, formally known as the Arqiva British Academy Television Awards, were hosted by Graham Norton at the Theatre Royal, Drury Lane, in London’s West End.
Last Tango In Halifax star Sarah Lancashire was named best supporting actress, beating her co-star Nicola Walker in the process before Gogglebox won the Bafta for reality and constructed factual.
BBC Three’s Him And Her: The Wedding was named best sitcom - one of two wins for the channel that is being axed by the corporation along with In The Flesh which won the Bafta for mini-series.
Blind Date presenter Cilla Black picked up the Bafta Special award from her fellow scouser Paul O’Grady and got a standing ovation.
She said: “I’d like to say a very big thank you to Bafta for this special award. You know I’ve led a charmed life, I really have, I’ve worked with incredible people.”
The singer and presenter dedicated her win to the “viewers” who she said had made her “feel so welcome over the years”.
The IT Crowd picked up two wins with the Bafta for female performance in a comedy programme going to Katherine Parkinson and her co-star Richard Ayoade getting the Bafta for male performance in a comedy programme.
Breaking Bad was named best international show and the Bafta for single drama went to Channel 4’s Complicit - about the torture of terrorism suspects.
Channel 4’s Bedlam won the Bafta for factual series, before the Radio Times Audience Award - voted for by the public - went to Doctor Who: Day Of The Doctor.
Other early awards included the sport and live event Bafta which went to Sky Sports’ coverage of day five of the first Ashes test and the current affairs gong which went to Channel 4’s Dispatches Syria: Across The Lines.
The Murder Trial was named best single documentary and ITV News At Ten won the award for news coverage for its work on the murder of soldier Lee Rigby in Woolwich.
The award for best soap went to Coronation Street, before Ross Kemp presented the Bafta for specialist factual to Sir David Attenborough’s Natural History Museum Alive 3D.
The award for comedy and comedy entertainment programme went to A League Of Their Own beating Norton’s chat show in the process.