The Pats opened the scoring in the first when Cameron Hebig picked up his sixth of the post-season.
In the second, Swift Current scored two straight to lead 3-1 but Robbie Holmes drew Regina to within a goal and it was 3-2 heading to the third.
In the final frame, the Pats outshot Swift Current 14-4 but could not find the tying goal and fell 3-2.
Regina outshot the Broncos 38-21, the powerplay went 0/1 and Regina did not have a penalty kill.
Sam Steel led the offence with two assists while Ryan Kubic made 18 saves.
The Broncos now advance to Round 2 and will play the winner of Moose Jaw-Prince Albert. Game 7 in that series goes Tuesday night at Mosaic Place.
The Pats will now wait 46 days before their next game, the opener of the 2018 Memorial Cup in Regina when the host team takes on the OHL champions on Friday, May 18.
(With files from Phil Andrews/ReginaPats.com)