The union's Unfair Labour Practice complaint, filed to the US National Labor Relations Board (NLRB), alleges that the company made "unilateral changes to terms and conditions of employment, without providing notice to the union or the opportunity to bargain, by utilising AI-generated voices to replace bargaining unit work".
It said this amounted to a failure by the employer to "bargain in good faith".
In the game, players can fight against Darth Vader, recruit the character to their team and interact with it.
"Strategise with the ultimate Sith Lord. With your voice. And he'll respond back using conversational AI," the company wrote in an announcement on Friday.
It said its AI voice recreation of the Star Wars villain would not have been possible, nor "nearly as sinister" without the agreement of the estate of James Earl Jones - the late actor who voiced Darth Vader in the film franchise.
An attached statement from the actor's family said he "felt that the voice of Darth Vader was inseparable from the story of Star Wars" and "always wanted fans of all ages to continue to experience it".
"We hope that this collaboration with Fortnite will allow both longtime fans of Darth Vader and newer generations to share in the enjoyment of this iconic character," they added.
Sag-Aftra said it welcomed its members and their estate exercising their right to control how digital replicas are used - but wanted to be involved so actors more widely could benefit.
"We must protect our right to bargain terms and conditions around uses of voice that replace the work of our members, including those who previously did the work of matching Darth Vader's iconic rhythm and tone in video games," it said.
The complaint comes against the backdrop of video game and voice actors growing increasingly concerned over the use of AI to reduce their work or replace them altogether.
Hollywood actors such as Scarlett Johansson have also highlighted concerns over how their digital likeness could be manipulated using generative AI.
Sag-Aftra, which represents actors across gaming, film, television and radio, has been continuing to try and reach agreements on terms around video game voice actors rights since strikes began last year.
Voice actors connected to the SAG-Aftra union have been refusing to work with companies like Activision, Disney, Warner Bros and EA since July over issues such as the use of generative artificial intelligence.
These included assurances over the use of digital replicas in video games.
There are also worries the AI-powered tech could be misused.
Wired reported that Fortnite players were already finding ways to make Darth Vader's character swear in voice chats.