Dallas (10-2-13) played its 16th straight game without a loss and moved closer to its first playoff berth in three years. The 13 ties for Dallas leads the league.
Ilija Stolica gave New England a 2-0 lead in the 66th minute and the Revolution appeared likely to pull off what would been a large upset.
David Ferreira converted a penalty kick in the 80th minute to give Dallas a chance and Cunningham came through with his ninth goal of the season to tie it.
New England fell to 7-14-3 and is in seventh place in the Eastern Conference.