The Union won their first home game and extended their unbeaten streak to three while goalkeeper Andre Blake earned the shutout.
The Red Bulls had won their previous two games by scores of 2-0.
In the third minute, Blake collided with teammate Jack Elliott in the air. Both players stayed down for several minutes before play resumed.
The Union went ahead 1-0 in the ninth minute when Burke hit a left-footed shot in the far corner of the net while falling to the ground.
New York earned consecutive corner kicks in the 18th and 19th minutes and continued its aggressive play.
In the 37th minute, the Red Bulls' Fabio received a cross in the box and headed the shot over the net.
The Union earned a free kick in the 42nd minute when Kai Wagner was fouled about 30 yards from the net. The cross was cleared but led to a pair of corner kicks.
Philadelphia's Sergio Santos fired a left-footed shot just over the net in the 54th minute, but it was slightly deflected for a corner kick.
Olivier Mbaizo took a solid shot in the 63rd minute, but Red Bulls goalkeeper Carlos Coronel was in position to make the save.
The Red Bulls then regained some offensive momentum and began to create more scoring chances. In the 70th minute, a corner kick was earned, but the Union defended it well and turned it into a goal kick.
Blake went down awkwardly again in the 88th minute after a diving save and took a couple minutes to recover and get back up.
The game became quite chippy in the first of three minutes of stoppage time with both teams pushing and shoving. New York's Dru Yearwood picked up a second yellow card and was sent off.