Jett Woo and Brett Howden had third-period goals for the Warriors, who were down 2-0 after the first two frames. Brayden Burke and Kale Clague had two assists apiece.
Moose Jaw opened the game trailing 3-2 in the best-of-seven series.
Swift Current's Matteo Gennaro scored the game's first goal on a power play 13:43 into the second period and Tyler Steenbergen added his ninth of the playoffs early in the third.
Broncos goaltender Stuart Skinner made 35 saves.
The Warriors host Game 7 Monday night.
The winner of the series will play the Lethbridge Hurricanes in the league semifinals. Lethbridge advanced by beating the Brandon Wheat Kings in five games earlier this week.