Trailing 34-22 with less than 6 minutes to go, Manning drove the Giants to two scores in the closing minutes, including an 8-yard TD pass to Jake Ballard with 3:14 remaining, cutting the deficit to five points.
After forcing the Cowboys to punt on the next possession, the Giants struck again.
Manning (27-of-47, two TDs, one interception) hit Ballard with a pair of long passes to set up Brandon Jacobs' go-ahead 1-yard TD run with 45 seconds left.
D.J. Ware added a 2-point conversion run after the score.
Dan Bailey's final bid for score-tying field goal was blocked by Jason Pierre-Paul and the Giants (7-6) remained alive in the NFC wild card hunt with the comeback victory.
Hakeem Nicks made seven catches for 154 yards and Victor Cruz also hauled down seven passes for 83 yards. Jacobs ran for 101 yards on 19 carries for New York.
Tony Romo completed 21-of-31 passes for 321 yards and four touchdowns for Dallas (7-6), which lost its second straight.