The HBO fantasy drama triumphed in three categories at Sunday's ceremony, including outstanding drama series.
The show's total number of awards now stands at 38, which means it has beaten Frasier's previous record of 37.
This year's British winners included Dame Maggie Smith, who won outstanding supporting actress in a drama series for her role in Downton Abbey.
It is the third time Dame Maggie has won an Emmy for her portrayal of Violet Crawley, but she has never attended the ceremony in person.
Host Jimmy Kimmel joked: "We're not mailing this to her. Maggie, if you want this, it will be in the lost and found."
BBC One's Sherlock was named best made for TV movie for its special episode The Abominable Bride.
Accepting the award, Sherlock creator Steven Moffatt made reference to the The Great British Bake off, currently shown on BBC One but bought by Channel 4 last week. He said: "Thank you to the BBC who we love above all bakery. British people will get that."
Susanne Bier, the Danish director of the BBC's spy thriller The Night Manager, won for best directing in a limited series.
Bier, who was the only female nominee in the category, told BBC Breakfast: "This is such a traditional men's world, and I hope the fact a woman director has won this prestigious prize is going to mean that more non-conventional series and movies are going to be directed by women."
John Oliver won best variety talk series for his HBO series Last Week Tonight, beating fellow British nominee James Corden, who was nominated for The Late Late Show.
Veep won the outstanding comedy series award for the second year in a row, while its star Julia Louis-Dreyfus won outstanding lead comedy actress for the fifth time.
In her acceptance speech, Louis-Dreyfus said: "Our show started out as political satire but now feels like a sobering documentary." She dedicated her award to her father, who died on Friday.
The outstanding comedy actor gong went to Jeffrey Tambor, who plays a transgender woman in the Amazon series Transparent, for the second year in a row.
Accepting the award, Tambor encouraged the industry to offer more opportunities to the trans community.
"I would not be unhappy were I the last cisgender male to play a transgender female on television. We have work to do," he said.
Rami Malek won outstanding lead actor in a drama series for his role in Mr Robot - the first time an actor from an ethnic minority has won the award since 1998.
Tatiana Maslany from Orphan Black won the outstanding leading actress in a drama prize.
Other big winners of the night included The People vs OJ Simpson - a dramatisation of the OJ Simpson trial of the 1990s, which was broadcast on BBC Two in the UK.
The show won five trophies including outstanding limited series plus acting gongs for Sarah Paulson, Sterling K Brown and Courtney B Vance.
Kate McKinnon, who recently starred in the all-female Ghostbusters reboot, won outstanding supporting actress in a comedy series for her performances in Saturday Night Live.
Netflix series Master of None scored its first Emmy award, winning outstanding writing for a comedy series.
The show's co-writer Alan Yang called for better Asian representation on screen in his acceptance speech.
Elsewhere, The Voice took home the award for outstanding reality competition series, beating Project Runway and Dancing With The Stars.
The 38 Emmys won by Game of Thrones includes the nine trophies the series picked up at last weekend's Creative Arts Emmys.
The HBO series is now the most decorated drama series in Emmy awards' history - the overall record is held by Saturday Night Live, which has won 44.
Kimmel made several references to diversity during the ceremony in light of the "Oscars So White" controversy earlier this year.
"Here in Hollywood the only thing we value more than diversity is congratulating ourselves on how much we value diversity," he said in his opening monologue.
"The Emmys are so diverse this year the Oscars are now telling people we're one of their closest friends."
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Game of Thrones has broken the record for the highest number of Emmy Awards won by any fictional series.