Six players recorded a multi-point game for the Oil Kings, who lead the best-of-seven second-round matchup 3-0.
Cody Corbett scored once and set up two more for Edmonton, which has outscored the Tigers 17-3 in the series, while St. Croix tacked on an assist for his third three-point performance of the playoffs.
Trevor Cheek and Travis Ewanyk also scored for the Oil Kings as David Musil and Stephane Legault each chipped in with two assists.
Legault leads all Edmonton skaters with 13 points in eight contests while St. Croix leads his club with seven goals.
Curtis Valk and Logan McVeigh scored for Medicine Hat, which found itself trailing 7-0 before getting on the board early in the third period.
Laurent Brossoit made 25 saves for the win as Cam Lanigan took the loss after allowing four goals on 27 shots before getting the hook early in the second period. Marek Langhamer replaced Lanigan and stopped 25-of-30 shots in 38 minutes of relief.
Brossoit, who has three shutouts so far in the playoffs, gave up two goals for just the second time in eight post-season starts. He leads all goaltenders with a 0.85 goals-against average and .966 save percentage.
Edmonton went 3 for 7 on the power play while Medicine Hat scored once on eight chances with the man advantage.
The Tigers will look to avoid elimination on Wednesday when they host the Oil Kings at the Medicine Hat Arena.