Cox took home the award in Best Performance by an Actor in a Television Series - Drama category at the award gala held on Sunday night here.
Cox beat out Kit Harrington ("Game of Thrones"), Rami Malek ("Mr. Robot"), Tobias Menzies ("The Crown") and Billy Porter ("Pose").
During his acceptance speech, Cox opened up about the significance of taking home the award and even apologised to his fellow nominees for winning.
"I'm sorry. I mean, this kind of event does your head in," began Cox.
"First of all, I want to apologise to my fellow nominees for winning this. I mean, I'm sorry. And that's all I can say. I'm sorry, but I never expected this. And I've been in this business, while next year, I've been in this business for 60 years, and I thought this would never happen to me. I'm a wee bit shocked."
The 73-year-old also gave a shout-out to the "extraordinary bunch of people" working on "Succession", including showrunner Jesse Armstrong.
He described Armstrong as "an incredibly modest man", "a talent" and "a gift", adding "if it wasn't for you, I wouldn't have had this."
"Succession", which airs in India on Star World, follows the lives of the Roy family as they contemplate their future once their aging father and media mogul Logan Roy (essayed by Cox) begins to step back from the media and entertainment conglomerate.
In an earlier interview to , Cox revealed that the makers initially planned to kill off his character.
"The original idea was to kill my character off at the end of the first season. But then they decided to keep me on...They decided that I will have a stroke and then the whole question of succession will come under scrutiny," he said.
"The show is a modern version of 'Game Of Thrones' except that we don't chop people's heads off."
( With inputs from IANS )