Assassination of Gianni Versace's Darren Criss won outstanding performance by a leading actor in a miniseries or television movie at Sunday's SAG Awards

Darren Criss took home the statue for outstanding performance by a leading actor in a miniseries or television movie at Sunday night’s Screen Actors Guild Awards for his role in The Assassination of Gianni Versace: American Crime Story.

The recently engaged Criss, who played serial killer Andrew Cunanan, used his acceptance speech to reflect on playing the man who murdered the legendary fashion designer.

“I find it appropriate that this statuette is kind of choosing between which masks to use,” Criss, 31, said at the Schine Auditorium. “The character that I played, Andrew Cunanan, unfortunately used masks to destroy things. As an actor my goal was to hopefully create something positive, to use masks to create a positive change.”

Credit: Michael Buckner/Variety/REX/Shutterstock

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He also had a message for the families affected by his character’s real-life killings.

“For any of the family and friends of those that are still affected by the destruction that he brought, I hope you know that our goal was not to make a spectacle of their tragedy but to create a positive dialogue about social issues and bring to justice things that were in the shadows,” Criss said.

Criss as Cunanan.
| Credit: Ray Mickshaw/FX

For the same part, Criss won the award for best actor in a limited series at the 2019 Golden Globes. He also previously won an Emmy last year for his portrayal of Andrew Cunanan in the FX series.

Criss beat out fellow nominees Antonio Banderas (Genius: Picasso), Daniel Brühl (The Alienist), Benedict Cumberbatch (Patrick Melrose) and Hugh Grant (A Very English Scandal).

After getting cast in the true crime anthology, which was a far cry from his past role as preppy singer Blaine Anderson on Glee, he previously told PEOPLE, “I had a friend tell me when I got the part, ‘You’re playing the gay boogeyman.’ ”

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When it came to playing Cunanan, Criss wanted to make sure he was portraying all of the complex aspects of his character, which included everything from killing in cold blood to singing Laura Branigan’s “Gloria” at the top of his lungs in his car.

“This was the first time I was in a position of any type of leadership on a show,” Criss tells PEOPLE. “For this particular project, it was a relief to be able to set the tone. When you have so much great material and you have a crew that’s working long days and everybody is away from their families and you’re repeating very gruesome things over and over again, it’s nice to know that I’m in the position to crack a few jokes and keep people light-hearted and help them disassociate from all the darkness that we were in.”

The 25th annual Screen Actors Guild Awards, hosted by Megan Mullally, are broadcasting live from Los Angeles on both TNT and TBS.