Lupita Nyong'o was also honoured at the ceremony, while Bryan Cranston and Maggie Smith triumphed in the television categories.
Dallas Buyer's Club celebrated a double win, with cast members McConaughey and Leto picking up the awards for 'Male Actor in a Leading Role' and 'Male Actor in a Supporting role' respectively.
Upon receiving his award, McConaughey paid tribute to his fellow nominees, saying: "There were so many fierce performances this years. I mean the choices in the characters that some of these people have made really shines a great light on this bull-ride that we take called acting."
The star, who shed 40lbs to play AIDS patient Ron Woodruff in the movie, described the "magic place" where an actor connects with a role to such an extent that "everything you see, smell and touch and observe is coming through that character."
Meanwhile, McConaughey's co-star Leto dedicated his award to all those who have lost their lives to AIDS or are currently living with the disease. He added: "I'm so proud that I've been able to glimpse the world through your eyes."
Leto also thanked his mother, who accompanied him to the ceremony, describing her as "the hottest date in town".
The major movie awards in the female categories went to Cate Blanchett - who won her third SAG Award for her lead role in Blue Jasmine - and Lupita Nyong'o, who triumphed over fellow nominees including Jennifer Lawrence, Julia Roberts and Oprah Winfrey to take home a trophy for her part in 12 Years a Slave.
During her acceptance speech, Nyong'o thanked Solomon Northup - the man whose memoir provides the basis of the movie - as well as her "amazing" drama school and her family who have "kept it real".
Breaking Bad's Cranston won the 'Male Actor in a Drama Series' accolade for the second year running for his role as Walter White. The actor broke into song upon winning the award, before describing all the "crappy" jobs he had before turning to acting.
The show's cast had another reason to celebrate as they also won 'Ensemble in a Drama Series'. Cranston declared: "What a way to go out in style!"
Elsewhere, Smith won 'Female Actor in a Drama Series' for her portrayal of Lady Violet Crawley in Downton Abbey, while Veep actress Julia Louis-Dreyfuss took home 'Female Actor in a Comedy Series.
The Modern Family cast was awarded 'Ensemble in a Comedy Series', while their very own Ty Burrell won 'Male Actor in a Comedy Series'.
Michael Douglas and Helen Mirren were both recognised in the 'Television Movie or Miniseries' categories, for their roles in Behind the Candelabra and Phil Spector respectively.
American Hustle won 'Cast in a Motion Picture', while Lone Survivor and Game of Thrones were celebrated for their stunt ensembles.
81-year-old West Side Story actress Rita Moreno was honoured with the 'Lifetime Achievement' accolade.
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