Colten Mayor and Adam Rossignol also scored for the Pats (8-14-2) while Luke Fenske added three assists.
Tyler Stahl and Jordan Fransoo responded for the Royals (11-11-0).
The highlight of the opening period was a spirited fight between Victoria defenceman Fransoo and Regina winger Dyson Stevenson.
The Royals opened the scoring early in the second when Stahl converted a perfect pass from Logan Nelson on the power play.
The Pats tied the game on a power play of their own when Mayor deflected a point shot high past Royals' goaltender Patrik Polivka.
Regina went ahead 2-1 in the last minute of the period when Rossignol banged one in from between Polivka's pads.
The Royals pushed hard in the third period with several quality chances and with just over two minutes left in regulation time, a big rebound was picked up by Fransoo and snapped high over Teagan Sacher's right shoulder.
Regina responded with just 25 seconds left to win the game when Klimchuk deked around the Victoria defence and snapped a low shot past Polivka.
Sacher made 32 saves for the win as Polivka stopped 18 shots in a losing cause.
Regina went 0-for-3 on the power play while Victoria went 1-for-6.