TORONTO -- Their postseason hopes have been fading for a while, but the Red Sox are playing hard until the end, and staving off elimination for as long as they can.

Nobody is exemplifying that as much as manager Alex Cora, who ran through his entire bench, gave up the DH and emptied out the seven relievers he had available -- all in an effort to keep his team alive for at least another day.

Down by three runs after six innings on Tuesday night at Rogers Centre, the Sox stormed back to force extras before rallying for three in the top of the 10th to down the Blue Jays, 6-5.

“It was a great team win,” said starter Brayan Bello, who went four innings and gave up two runs. “I was on the bench giving some cheers, but the guys in the bullpen did an awesome job. It was a nice team effort tonight.”

With four games left in the season, the Sox trail the Royals and Tigers by 3 1/2 games for the third American League Wild Card spot. Any path for the Red Sox advancing past Sunday would involve winning their final four games and ending the regular season on an eight-game winning streak. But they would still need help for that to mean anything.

“We don’t back down from a fight,” said second baseman Vaughn Grissom. “We’re just gonna keep our head down to push and see what happens.”

The Red Sox (80-78) extended their winning streak to four games, the longest the club has had since winning five in a row from June 30-July 5.

The most recent win required participation from 22 of the 28 players on the roster.

“It was just fun,” said Cora. “We put ourselves in this position, and obviously it's not a perfect one, but for us, it's just going out there to try to make it count. And the boys did an amazing job fighting all the way to the end. We used everybody, and it was one of the most gratifying wins since I've been here.”

Trevor Story banged a double off the wall in left for the go-ahead hit in the 10th, before Grissom tacked on some insurance with an RBI single, his third hit of the night.

It was the most impactful performance of a frustrating season for Grissom, who battled hamstring issues for much of the year and rebuilt his game and his confidence for over a month at Triple-A Worcester before earning his way back to the club on Sept. 20.

“It was sweet,” said Grissom. “Any time you can give your team a chance to get ahead like that late in the game, it’s sick. Obviously, I missed a lot of time, and it's been good to just get out there and compete with the guys and help put us in a good spot.”

“Vaughn did a good job staying on pitches. The walk in the middle of the game was an outstanding at-bat. [Wilyer] Abreu put together another great at-bat,” said Cora. “Trevor strikes out against the lefty, and then he got a slider in the zone [in the 10th] and put a great swing on it. Everybody contributed. Everybody. Mickey [Gasper] with the baserunning play after we pinch-ran for [Triston] Casas. Casas with the at-bats. We can go on and on and on today. That was great.”

What led Cora to be so aggressive with his roster?

“Trying to win the game,” he said. “You cannot do that every time. Sometimes I wonder why not to be honest with you, because we're in the business of winning games, but we understand that’s not possible, right? And even today, we had to stay away from [relievers] [Josh Winckowski] and [Greg] Weissert in this game, and the boys did an amazing job.”

Though a trip to the postseason is still unlikely, the Red Sox will show up for Game No. 159 on Wednesday night still breathing.

It was Bello who let out a big cheer in the clubhouse after the game when the Astros completed their win over the Mariners. In addition to trailing the Royals and Tigers, the Sox also trail the Mariners by one game, and the Twins by 1 1/2 games. Boston owns the tiebreaker against the Royals and Mariners.

“It would be great if we make it, it would be the first playoffs in my career,” said Bello. “So obviously we’re pushing for the teams ahead of us to lose and we’ll see what happens.”