Ben Roethlisberger completed 20 of 25 passes for 191 yards with a touchdown and an interception. Le'Veon Bell added 117 yards of total offense.
Andy Dalton completed 25 of 44 for 230 yards and two second-half touchdowns, but Cincinnati (9-5) never recovered after getting staggered early at frigid Heinz Field.
Looking to move closer to a third straight playoff berth, the Bengals were undone by a series of miscues on special teams.
Bell scored two plays after Cincinnati punter Kevin Huber dropped a snap while standing near the goal line after the Bengals' first possession. Pittsburgh's Will Allen dropped Huber at the 1 and Bell bulled into the end zone on second down for his sixth touchdown of the season.
Cincinnati's issues continued on the ensuing kickoff. Officials ruled Cedric Peerman called for a fair catch at the Cincinnati 9, wiping out a lengthy return. The Bengals eventually punted, setting up the Steelers with a short field. Pittsburgh went 47 yards in eight plays, with Roethlisberger buying enough time for Brown to get open for a 12-yard TD strike to make it 14-0.
Brown's early work wasn't done. Cincinnati punted yet again and Brown took it 67 yards right up the middle, breaking a tackle at midfield on his way to the second punt return for a score in his four-year career for Pittsburgh's third touchdown of the quarter.
The Steelers, who need to win out to avoid their first losing season since 2003, had been outscored 77-43 in the first quarter coming in.