Brown's boot came after Bengals safety Reggie Nelson intercepted a Ben Roethlisberger pass with 14 seconds remaining in regulation and Andy Dalton moved Cincinnati into field goal range with a 21-yard completion to A.J. Green.
With the victory, the Bengals (9-6) made the playoffs for a second straight season for the first time since 1981-82.
The Steelers (7-8), however, were knocked out of playoff contention with their third straight loss and fifth in six games.
Roethlisberger finished just 14-of-28 through the air for 220 yards and one touchdown but was picked off twice -- including one interception that was returned for a touchdown by Leon Hall.