Foy won the best actress in a drama series for her role as Queen Elizabeth II in Netflix's royal epic The Crown.
Game of Thrones won two prizes - best drama series and supporting actor for Peter Dinklage - while The Marvelous Mrs Maisel scooped five comedy awards.
However, two hotly-tipped shows - Atlanta and The Handmaid's Tale - failed to replicate their successes from last year and went home empty-handed.
Elisabeth Moss had been the pre-show favourite to win best drama actress for the second season of The Handmaid's Tale, but was beaten by Foy.
In her acceptance speech, the British star described her time on The Crown as "the most extraordinary two-and-a-half years of my life".
She said: "I was given a role I never thought I would ever get a chance to play, and I met people who I will love for ever and ever.
"And the show goes on, which makes me so proud. So I dedicate this to the next cast, the next generation, and I also dedicate this to [her co-star] Matt Smith."
Olivia Colman is due to take over the role from Foy as the show enters its third season.
Newton was awarded best supporting actress in a drama series for her role in Westworld.
"I don't even believe in God but I'm going to thank her tonight," the actress said as she took to the stage to collect her trophy.
Rhys picked up best leading actor in a drama series for his role in The Americans.
Set in suburban Washington DC shortly after Ronald Reagan is elected president, the series tells the story of two KGB spies in an arranged marriage who are posing as Americans.
"Parts like these come along so rarely. I will forever be in your debt," he told series creator Joe Weisberg in his speech.
Charlie Brooker, the British creator of Black Mirror and his co-writer William Bridges won best writing for a limited series.
John Oliver added to the British success at the ceremony, picking up the prize for best variety talk series for Last Week Tonight with John Oliver.
The Marvellous Mrs Maisel was one of the night's other biggest winners - adding five trophies to the four it won last week at the Creative Arts Emmys.
The series took home best writing, directing, lead actress and supporting actress in the comedy categories, as well as one of the night's big prizes - best comedy series.
The Assassination of Gianni Versace was also one of Monday's big winners - scooping best limited series, best directing for a limited series, and best lead actor in a limited series for Darren Criss.
Henry Winkler won his first Emmy - supporting actor in a comedy series for his role in Barry - 42 years after he was first nominated for playing The Fonz in Happy Days.
"Skip Brittenham said to me a long time ago, 'If you stay at the table long enough, the chips come to you,' and tonight I got to clear the table," he said in his acceptance speech.
Jeff Daniels took home the prize for best supporting actor in a limited series or a television movie for his role in Netflix's Godless - described as a feminist western.
In addition to his family and co-workers, Daniels thanked the streaming service for "letting artists be artists".
He added: "Little tip for you young actors - when they call and say 'Can you ride a horse?' don't lie. You will find on day one that you're in the Kentucky Derby."
Put together with the Creative Arts Emmys handed out last week, this year's awards tally saw Netflix and HBO tied, with 23 awards each.
But one of the night's most memorable moments came from Glen Weiss, who proposed to his girlfriend from the stage during his speech.
While accepting outstanding directing for a variety special for The Oscars, he spoke to his partner Jan Svendsen, who was sitting in the audience.
"You wonder why I don't like to call you my girlfriend? Because I want to call you my wife," he said as the crowd began cheering.
Svendsen, who looked shocked and delighted, then made her way up to the stage to accept his proposal.
Weiss said he was giving her the ring "that my dad put on my mom's finger 67 years ago". His proposal came just two weeks after the death of his mother, which he also referred to in his speech.
Rhys joked about the moment later in the night, telling the audience as he collected his award that his girlfriend had whispered to him: "If you propose to me I will punch you clean in the mouth."
The ceremony took place at the Microsoft Theatre in Los Angeles, with Kenan Thompson and Kate McKinnon kicking off proceedings.
"We're celebrating the fact that this year's Emmy Awards has the most diverse group of nominees in Emmy history," Thompson said.
"One step closer to a black Sheldon," he joked - referring to The Big Bang Theory's most famous character.
The pair then embarked on a parody song called We Solved It - suggesting the Emmys had single-handedly fixed Hollywood's lack of diversity - which featured cameo appearances from Sandra Oh and John Legend.
Ceremony hosts Colin Jost and Michael Che then took to the stage - referencing the fact that streaming service Netflix received the most nods this year in their opening monologue.
"NBC has the most nominations of any broadcast network," Che said, adding: "Which is kind of like being the sexiest person on life support".
Because the Emmys have a total of 122 categories, the less high profile awards were given out last weekend at the Creative Arts Emmys, with 26 awarded on Monday night.
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