Leading by one goal with 4:24 left to play, one of their top penalty-killing defenceman, Dakota Mermis, was sent off for kneeing, followed a minute later by teammate Cole Cassels for delay of game.
But the Generals managed to kill off the two-man advantage and preserve the 2-1 victory over the Western Hockey league champion Kelowna Rockets on Tuesday night.
"The clock couldn't go fast enough, but fortunately we have a really deep team and the guys stepped up and got the job done," said Mermis.
Cassels and Tobias Lindberg scored in the second period and the Generals, who dominated the opening two periods, held on to finish the round-robin portion of the four-team tournament at 3-0. That gave them direct entry into Sunday's final.
"We're certainly excited but the work's not over," said Cassels. "We've been waiting for that Sunday game all year. It'll be the biggest moment of our lives."
Gage Quinney, with his fourth of the event, scored for the Rockets (1-2), whose fate will be determined by the outcome of a game Wednesday night between the Quebec champion Rimouski Oceanic (0-2) and the host Quebec Remparts (1-1).
The final shot tally was 29-21 for Oshawa, but Kelowna had only six shots by the mid-point of the second frame, with Michael McCarron doing a masterful job of neutralizing Rockets star Leon Draisaitl.
Kelowna, which had the only three power plays of the game, only made it close when the Generals sat back in the third to protect the lead.
The Rockets had a handful of glittering chances during their two-man advantage, but either missed open nets or had their shots absorbed by goalie Ken Appleby.
"We didn't bear down and we didn't get the result we needed," said Rockets captain Madison Bowey. "But we're not out of it.
"We have to climb back, but it's not over."
The Generals had their forechecking and cycling game going from the outset and had a 12-5 shot advantage in the first, but couldn't find a goal. McCarron swung wide and whipped a shot off the far post in the dying seconds but it stayed out.
Cassels took a feed from Mermis to score on a quick shot from the left side 4:15 into the second.
Lindberg opted to keep the puck on a 2 on 1, make a toe-drag and beat Jackson Whistle between the pads at 17:16.
Only 55 seconds later, however, a perfectly executed rush from Bowey to Tyson Baillie to Quinney, who scored on a backhand, put Kelowna back in the game.
"We had a few chances but we looked nervous a bit," said Rockets coach Dan Lambert, whose squad had fired seven pucks past two Rimouski goalies on Monday night. "We were squeezing our sticks too tight."
The Remparts can book a spot as the "home" team in the semifinal if they defeat Rimouski, the team that edged them in overtime in Game 7 of the Quebec Major Junior Hockey league final.
The start of the game was delayed by an on-ice tribute and banner raising ceremony for former goaltending great Patrick Roy, who was owner, GM and coach of the Quebec Remparts. He got a long standing ovation.