The Eagles had grabbed the lead when quarterback Nick Foles found Zach Ertz on a 3-yard touchdown pass with 4:54 left, but the Saints quickly responded.
The game-winning boot was set up when Drew Brees made good on two third-down plays with a pair of quarterback sneaks during a last-chance drive as time wound down.
Graham -- who replaced Garrett Hartley late in the season -- easily nailed the ensuing kick, giving the Saints their first road playoff win in six tries. They next face the top-seeded Seahawks in Seattle.
Brees converted 20-of-30 pass for 250 yards with a touchdown and two interceptions.
Mark Ingram led the Saints rushing corps with 97 yards and a score on 18 times carries.
Foles finished 23-of-33 for 195 passing yards and two touchdowns for Philadelphia.