Cummings, who struck a late equaliser in the first leg in Houston, netted in the 104th minute as the Dynamo upset the Eastern Conference's top seeds in the play-offs.
Bradley Wright-Phillips had given the Red Bulls the lead, only for his opener to be cancelled out by Brad Davis.
And it was left to Cummings to be the hero for the Dynamo, leading them to a 4-3 aggregate win and seeing them into the Eastern Conference final.
Wright-Phillips put the Red Bulls ahead in New Jersey on 23 minutes, pouncing when Tally Hall spilled a cross.
But their lead only lasted 13 minutes before Davis hit back for Houston to level the game and the tie.
New York looked the more likely to find a winner but it was Houston who did so just a minute before half-time in extra-time.
Cummings flicked an effort towards goal and Luis Robles was too slow to get down to his left and unable to stop it crossing the line.
Awaiting the Dynamo in the conference final are Sporting Kansas City, who needed extra-time to beat the New England Revolution 3-1.
Sporting, who lost the first leg 2-1, levelled the tie on 41 minutes when Aurelien Collin struck from inside the area.
Dimitry Imbongo looked to have scored the winner for New England with 20 minutes remaining as he volleyed in a set-piece.
But Kansas City struck again just nine minutes later through Seth Sinovic, whose stunning left-footed effort found the bottom corner from the side of the area, before Claudio Bieler scored the winner with seven minutes left in extra-time.
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