After neither team could score in their first overtime possessions, the Bengals' Brandon Tate returned a Brian Moorman punt 29 yards to the Buffalo 33.
That set up Nugent's game-winning kick, which lifted Cincinnati (4-2) to its first win in Buffalo since 1985.
The Bengals' victory, however, only came after backup quarterback Thad Lewis had sparked the Bills (2-4) to a comeback late in regulation.
Making just his second career NFL start while subbing for injured EJ Manuel, Lewis tossed a pair of fourth-quarter touchdowns, including a 40-yard game-tying scoring strike to rookie Marquise Goodwin with 68 seconds left.
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