When Tim Cahill’s cultured last-ditch header floated over New England keeper Matt Reis and into the net, the Red Bulls had pulled out a 2-2 draw that clinched a playoff berth. Yes, they squandered a game they should have won, but they also pulled out a tie where a few months ago they would have lost.

Coughing up Lee Nguyen’s equalizing penalty kick in the 85th minute and Diego Fagundez’ goal a minute into added time put the Red Bulls on the verge of a loss. Cahill’s score seven minutes into added time pulled out the much-needed point to clinch a playoff bid. Their next goal is winning the Supporters’ Shield, given to the team who finishes the regular season with the most points.

“I thought my heart was going to give out. It’s a great moment for organization. But this is our initial goal. Now our goals get recalculated,’’ rookie coach Mike Petke said. “The second goal right now is the Supporters’ Shield, 100 percent. It would’ve been a lot more realistic if we’d gotten three points.’’

One didn’t have to go far to actually see the object of their desire, with the Supporters’ Shield visiting a bar El Pastor just across the footbridge in Newark. They have made the playoffs 10 of 11 years, but after being frustrated contenders, now they want to be champions.

At 53 points with two games left, the Red Bulls don’t control their destiny. Sporting KC (51 with a game in hand) can catch them if they win out, with Seattle and Real Salt — each tied at 51 points — playing Saturday night, the former with three games in hand and the latter with one.

But they have shown plenty of fight, with a 5-0-2 record since Petke’s practice field fight with Thierry Henry. They had to be separated that day, but the team has grown closer.

“At the end of the day we didn’t lose. A couple months ago we would’ve lost that one,’’ Henry said. “Somehow we found a third, fourth gear to make sure we were not going to give up. That’s something we have now.’’

They were on the verge of a collapse, losing their clean sheet in 83rd minute when CB Jamison Olave was called for a handball in the box on a Saer Sene cross and Nguyen buried the penalty kick. The Red Bulls almost lost the game when Fagundez picked off David Carney’s back-heel backpass and scored.

But after an injury to New England’s Andrew Farrell extended added time, Cahill got a pick from Henry and Peguy Luyindula, and lofted a long header that chipped Reis for his team-high 10th goal of the year.

“When I came here first it was a bit messy. … It was accepted some things that happened in games, and losing,’’ Cahill said. “Now it’s a team that plays for each other, to the end of every game.’’