Nugent missed a 51-yarder in the first half and an extra point in the second half.
The Redskins dominated the first half, outgaining the Bengals 226 to 82 in the first half, but that was only good for a 10-7 lead.
Cincinnati opened the third quarter with an 11-play, 75-yard drive, taking their first lead on Andy Dalton’s 15-yard touchdown pass to Tyler Eifert for the tight end’s first scoring catch since Week 17 last year.
Eifert finished with nine catches for 102 yards, while A.J. Green caught nine passes for 121 yards, including a 40-yarder at the Washington 7 that set up Hill’s go-ahead score in the fourth quarter.
Mike Nugent missed the extra point after the Eifert TD, keeping the score at 13-10, and the Bengals upped their advantage to 20-10 a few minutes later when Dalton ran in from 3-yards out.
Washington answered with a 23-yard touchdown pass from Kirk Cousins to Jordan Reed.
The Bengals appeared ready to make it a two-possession game again with a first down at the Redskins 20, but Washington linebacker Will Compton picked off Dalton at the 12 for Dalton’s first interception in 165 attempts, a franchise record.
The Redskins converted the mistake into an 88-yard drive that Kirk Cousins capped with a 33-yard touchdown pass to Jamison Crowder to go up 24-20.
Cousins finished with 458 passing yards with two touchdowns and an interception.
Dalton threw for 284 with one TD and one interception.