The loss snapped the Raiders 19-game winning streak.
Prince Albert (26-1-0-1) were up 2-1 late in the third period when Regnier scored with 1:44 left in regulation to send the game into overtime.
Andrew Fyten scored on the power play in the first period for the other Swift Current (5-21-1-1) goal.
Brett Leason and Brayden Pachal scored for the Raiders. Leason now has at least a point in 28 straight games.
Ian Scott made 23 saves in the loss.
The Raiders are back in action on Friday when they travel to Brandon to open a home-and-home weekend with the Wheat Kings.