Peterson, who scored the game's first points on a 7-yard reception on their opening drive, capped a 10-play, 73-yard scoring drive spanning 5:38 in the first quarter, marking the sixth time this season an opponent scored on its opening drive against the Giants' defense.
The Giants had opportunities in the first half, but as has been the case all season for them, they squandered them away. Eli Manning, who finished 30-of-47 for 316 yards, one touchdown and two interceptions, was also sacked seven times for the fourth time in his career as the offense failed to generate much.
Manning did connect with receiver Odell Beckham Jr. on a 44-yard deep ball to get into Washington territory. However, the excitement was short lived as on the next two plays, Saquon Barkley rushed for no gain and Manning was sacked for a 5-yard loss. The Giants ended up punting the ball away after reaching the Washington 39-yard line.
The Giants would have another chance to even the score, driving into the red zone on their next possession in the second quarter. However, Manning's pass, intended for Beckham on a slant, was picked off by safety D.J. Swearinger on what was the first of his two interceptions.
The Giants' only first-half points came courtesy of a 37-yard field goal by Aldrick Rosas with 1:19 left in the second quarter. His counterpart, Dustin Hopkins, missed a 41-yard field goal in the second quarter but went on to connect on attempts of 53 and 39 yards to give his team a 13-3 lead.
The Giants' defense, who dealt away cornerback Eli Apple and defensive tackle Damon Harrison, and who was missing inside linebacker Alec Ogletree (hamstring), battled hard and tried to keep the game within reach for as long as it could.
The Giants' defense came up with a huge turnover at the end of the third quarter when safety Landon Collins hit Peterson to force a fumble that linebacker Olivier Vernon scooped up and returned 43 yards to the Washington 39-yard line.
Predictably, the Giants' offense, which cannot get out of its own way, tossed the opportunity away, moving the ball a net total of seven yards on four plays, including a fourth-down attempt that resulted in an incomplete pass thrown to tight end Evan Engram, the ball sailing right through his hands.
After taking over on downs, Hopkins' second field goal of the game made it a two-score lead.
The Giants' offense got another golden opportunity in the fourth quarter thanks to a pair of defensive pass interference calls against Washington, the first a 46-yarder against safety Montae Nicholson followed by a 4-yard penalty by cornerback Fabian Moreau to give the Giants first-and-goal on the Washington 4-yard line.
Once again, the Giants were unable to punch it into the end zone, instead settling for a 21-yard field goal by Rosas which made it 13-6 with 4:11 left in the game.
Washington dealt what would become the final blow to the Giants on their ensuing possession in which Peterson took it to the house on a 64-yard touchdown run to make it a 20-6 affair.
The Giants did manage to record a garbage-time touchdown on Manning's two-yard touchdown pass to tight end Evan Engram to make it 20-13 with 17 seconds left in regulation. Washington recovered the ensuing onside kick.
NOTES: WR Odell Beckham Jr. finished with 136 receiving yards, the 34th time in his career and the fifth time this season he topped the century mark. Beckham also moved into fourth place on the Giants' all-time career reception list, passing Frank Gifford, Ike Hilliard and Jeremy Shockey on his way up the ladder. ... Giants right tackle Chad Wheeler suffered an ankle injury in the third quarter and was replaced by Brian Mihalik.