Lindpere scored the first and Thierry Henry added the second in the closing seconds as the Red Bulls advance to the conference semifinals, where they will meet the Galaxy.
In a deciding Wild Card game, the New York Red Bulls did just enough with 10 men to knock off Brek Shea and FC Dallas with a 1-0 victory at Pizza Hut Park on Wednesday.
"It was a bitter game for us," FCD coach Schellas Hyndman said. "I thought we did everything we possibly could. No excuses, they came in and beat us on our field."
Joel Lindpere scored the first goal in the second half and Thierry Henry capped off the night by walking in an easy one in the closing seconds of extra time.
It was an impressive performance for the Red Bulls down the stretch as they played without a man for the final 10 minutes plus injury time after Jan Gunnar Solli received a late red card.
New York moves on to face David Beckham, Landon Donovan and the Galaxy in the first leg of the conference semis at Red Bull Arena on Sunday before shifting the series to Hollywood.
Meanwhile, Dallas' season is cut short and questions will surely arise surrounding the futures of Shea and defender George John. Several European teams have been linked to both of the breakout players since last summer's transfer window.
While the Red Bulls controlled the end of the game, the beginning belonged to the home team.
"We came out very lively, a really good team chemistry, team spirit, to do everything we possibly could to get the result we wanted," Hyndman said.
Dallas put the pressure on New York from kickoff and nearly scored off two Red Bulls turnovers in the first 15 minutes.
In the eighth minute, Jackson had a clean path off a New York mishap that forced Rost to make a play. Moments later, a cross bounced off Red Bulls defender Tim Ream's leg to Jackson's feet for another open attempt.
"The first 20, 25 minutes we were very good," said Hyndman. "We put really good pressure on New York, but we just couldn't get on the end of services or get the shot on frame."
It took a half-hour before the Red Bulls finally got something going offensively.
It appeared as though New York striker Dane Richards, starting in place of the injured Luke Rodgers, was taken down in the box by Ugo Ihemelu, who was given a yellow card. But replays showed the ball was just outside the lines and nothing came of a free kick.
The start of the second half started slowly and stayed at that pace for most of the period, as both teams sat back and relied on the counterattack for chances. And that's exactly how New York eventually took the lead in the 61st minute.
Lindpere fired a long-range shot that bounced off Hartman. Richards scooped up the rebound and laid it off to substitute Mehdi Ballouchy, who found Lindpere with a no-look pass between two defenders in the box for the finish.
Dallas responded by taking control just as it did in the game's first few minutes. Shea became extremely aggressive and would have tied it up with about 15 minutes remaining had it not been for a desperation save by Ream.
Shea dribbled into the box from the left and had Rost beat, but the New York defender made it to the goalline to kick the ball out of play just in time.
New York played the last 10 minutes with just 10 men as Solli got a red card for two-footed tackle on Daniel Hernandez, who spent some time on the ground.
"I can understand the referee after seeing the replay," NY coach Hans Backe said. "I can understand why he's given the red card."
The referee added seven minutes of extra time to the game, giving Dallas plenty of time to take advantage of the shorthanded Red Bulls. But it would come up short.
The team that would strike last would be the Red Bulls off a fluke play that found Henry in a one-on-one situation. Hartman tried to stop the ball with his feet but watched it slip through his legs. Henry just tapped it in and the final whistle was sounded.
"Tonight, I can't say we had a bad game," Hyndman said. "We just ran out of time."
Allen Ramsey contributed reporting to this article