The Red Bulls played much of the match without Juan Pablo Angel, who came out of the game after getting hit in the face with a knee.
Playing on a windy, rainy night at home, New England's Shalrie Joseph hit Steve Ralston with a long pass, and Ralston passed across the penalty are to Twellman, who put the ball just past the outstretched hands of Red Bulls goalkeeper Jon Conway.
With the win, second-seeded New England (15-8-9) advances to its sixth straight Eastern Conference final.
The loss sends New York (12-12-8) to its sixth straight first-round playoff exit.
The Revs will be at home Thursday night for an Eastern Championship match against the Chicago Fire, who beat top-seeded D.C. United Thursday.