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The visitors broke the deadlock in the opening quarter when Kai Forbath converted a field goal from 35 yards, but the Giants were soon ahead courtesy of the first touchdown of the match.
Manning and Beckham combined for the first time in the game to produce the score, the quarterback finding the receiver on the right side of the field for a 10-yard touchdown.
Robert Griffin III was involved as the Redskins regained their advantage, throwing a short pass to Chris Thompson for a nine-yard score, but the quarterback then had a rushing touchdown of his own disallowed at the end of the first half.
Josh Brown restored parity for the Giants with a 32-yard field goal at the start of the third quarter, but Forbath added a kick of his own to restore the three-point lead.
The Giants moved back into the lead late in the third quarter, and it was the same duo that combined again as Manning found Beckham, who sprinted between two defenders for the touchdown.
Beckham made it a hat-trick in the fourth quarter, opening up an 11-point gap for his side with his third touchdown of the match.
That was how it stayed as the Giants held out to condemn the Redskins to a sixth consecutive defeat, improving their own record to 5-9 in the process.