Calgary's 36-30 victory over Saskatchewan in Sunday's West Division semifinal propels the Stampeders to the division final against the B.C. Lions. The victor in Vancouver next Sunday advances to the Grey Cup Nov. 25 in Toronto.
Tate hit Romby Bryant on a 68-yard touchdown pass with 20 seconds remaining in the semifinal. A two-point convert attempt failed, Stampeder kicker Rene Paredes booted the subsequent kickoff 90 yards and the visitors conceded a single point.
"I knew no one was going to catch him," Tate said. "I was just hoping the ball was going to be a good ball."
The Roughriders had taken a 30-29 lead just 32 seconds before Bryant's heroics on a Darian Durant 24-yard throw to Gregg Carr.
Tate says he didn't remember much after taking a late helmet-to-helmet hit from Saskatchewan's Tearrius George early in the second quarter.
"I don't even remember the first half," Tate said. "I got my bell rung. All I remember is the second half."
Tate rose to his feet slowly after the George hit. He was flattened later on the same drive by Shomari Williams, who was called for roughing the passer as George was.
Calgary's starting quarterback remained in the game and completed 22 of 36 pass attempts for 363 yards and two touchdown throws.
In his post-game comments, Stampeder coach and general manager John Hufnagel said he was unaware of Tate's mental state after the George hit. Tate insisted after the game that he wasn't concussed.
In the East Division, the Toronto Argonauts head to Montreal and meet the Alouettes in the division final Sunday. The Argos were 42-26 winners over the Edmonton Eskimos in the semifinal.
Calgary receiver Jabari Arthur scored his first career playoff touchdown and Calgary third-string quarterback Bo Levi Mitchell finished short-yardage plays for a pair of touchdowns. Paredes was good on attempts from 50 and 40 yards.
Stampeder running back Jon Cornish, a finalist for the CFL's most outstanding player award, finished with 109 rushing yards on 18 carries.
Kory Sheets and Carr each collected a pair of touchdown catches and Sandro DeAngelis added a field goal for the Roughriders.
Saskatchewan's Durant was 24-for-37 for 435 yards, four touchdown throws and two interceptions.
"Darian had a phenomenal game," Saskatchewan head coach Corey Chamblin said. "He gave us a chance to win that game.
"At the end we just had a defensive breakdown. We handed it away and that should never happen."
Saskatchewan was a post-season stumbling block for the Stampeders in recent years, with four playoff wins over Calgary in five seasons between 2006 and 2010, and another back in 1997.
"If they were banking on history, they had another thing coming," Calgary defensive back Keon Raymond said. "It also feels good because out here, they beat us in 2009 and 2010. I was a part of that team and it's heartbreaking.
"It's always good to know we can win against them at home. If we can get that monkey off our back, try and make a run, that's who you want to go through."
It was looking like another Roughrider post-season win at McMahon Stadium until Tate aired the ball out to Bryant.
The temperature at kickoff at McMahon was -2 C. Attendance was announced at 30,027.
The first and second halves ended with bunches of points — 14 with 52 seconds remaining in the second and 13 with 12 seconds to go in the first.
In the wild conclusion to the first half, Calgary's Fred Bennett ran 96 yards back for a rare defensive two-point convert after Raymond blocked a Saskatchewan convert attempt. A 50-yard field goal from Paredes on the last play of the half gave the hosts a 19-16 lead.
Hufnagel announced earlier in the week that Tate would be his playoff starter, even though the Baytown, Texas, quarterback was sidelined 14 regular-season games with a shoulder dislocation and subsequent surgery on his non-throwing arm.
Backup Kevin Glenn went 9-5 in Tate's absence. Hufnagel went with Tate because of his athleticism.
Despite his foggy mental state, he scrambled out of trouble to make plays, including a 17-yard pass to Maurice Price in the third quarter when Tate was surrounded by green and white jerseys.
The University of Iowa product went 5-for-5 in passing for 68 yards on that scoring drive in the third quarter to make it 26-16.
"He managed the game well," Hufnagel said. "You saw some things he can do, making some plays out of nothing. We all need to play better that week and Drew is one of them."
Calgary won some tight games this season en route to a 12-6 record and second in the division. Three ended with the opposition missing field goals on the last play. The Stampeders won another in overtime in Regina.
"How many games have we won within the last minute, clutch field goals and things like that?" Raymond asked. "That's the kind of team we are. A team of character.
"If there's time on the clock, we still have an opportunity and we showed that tonight."
The Roughrider season was one of high highs and low lows. They opened this season 3-0, lost five in a row, went 5-1 and then finished on a four-game losing streak to finish 8-10.
"It's a young team. We have to come back stronger next year," Chamblin said. "Sometimes those lessons turn you into a veteran team."