The forthcoming film, based on the Marvel Comics character of the same name, stars Johnson, 34, as Cassandra Webb, a Manhattan paramedic who can see into the future.
Sweeney, 26, plays Julia Carpenter, who later takes on the role of Spider-Woman in the comics.
The trailer offers viewers a first glimpse of Sweeney in the comic-accurate, black-and-white costume.
The cast also includes Isabela Merced, Emma Roberts, Adam Scott, Celeste O’Connor, Tahar Rahim, Mike Epps and Zosia Mamet.
Last year, Euphoria star Sweeney spoke about her first time meeting Johnson, who she’s a huge fan of.
“I was a little nervous,” Sweeney said during a red carpet interview.
“I have to say — I snuck 50 Shades of Grey in high school. My parents didn’t let me watch it and I watched it. So I’m a huge fan of her. She’s so hot.”
Sweeney denied ever speaking with Johnson about her nerves while on set together. When asked if she’d revealed her intimidation to her co-star, she said: “No! But, Dakota, you’re hot and I loved working with you.”
In the Marvel comics, Madame Web is portrayed as an elderly woman who suffers from a neurological condition known as myasthenia gravis. Her name originates from the life support machine that she is connected to, which resembles a spider’s web.
Her condition, however, allows her to develop powerful psychic abilities, which she uses to become a medium.
SJ Clarkson, who helmed Marvel’s Jessica Jones and The Defenders, directs Madame Web from a screenplay she co-wrote.
The film comes off the back of Sony’s success with Tom Holland’s hugely popular Spider-Man film series. The third instalment, Spider-Man: No Way Home, was the highest-grossing movie of 2021, earning $1.9bn worldwide.
Back in December, Sony Pictures chairman Tom Rothman gave fans a potentially disappointing insight into the status of a prospective new Spider-Man sequel.
Because the rights to Spider-Man are owned by Sony, the MCU adaptations involve complex business deals with Marvel Studios (a subsidiary of Disney).
Madame Web is scheduled for release on 14 February 2024.