The Browns jumped out to a 28-0 lead in the first quarter and held on after halftime to punch their ticket to the AFC divisional round next weekend. The franchise's last postseason victory came against the New England Patriots on Jan. 1, 1995, when current Patriots head coach Bill Belichick was in the same position with the Browns.
Cleveland (12-5) will play the top-seeded Kansas City Chiefs in the next round for a chance to reach the AFC Championship Game.
"We believed," Browns quarterback Baker Mayfield said. "People on the outside don't matter to us. It's on the inside and we believed the whole time, and that's all that matters."
Playing without head coach Kevin Stefanski, star offensive lineman Joel Bitonio and top cornerback Denzel Ward due to COVID-19, the Browns quickly went up 14-0 after two costly turnovers by the Steelers.
Browns running back Kareem Hunt pushed his team's lead to 28-0 after two rushing touchdowns late in the opening quarter. Cleveland then scored again before halftime, as the Browns raced to the largest first half by a road team in NFL postseason history.
The Steelers (12-5) managed to convert a 49-yard field goal before the half to trim their deficit to 35-10. Pittsburgh then scored twice in the third to make it 35-23.
Facing fourth-and-1 at their own 46-yard line, Steelers head coach Mike Tomlin opted to punt. That decision ultimately backfired after Pro Bowl running back Nick Chubb took a screen pass 40 yards for a touchdown to give the Browns an insurmountable 42-23 lead.
Steelers rookie receiver Chase Claypool scored two touchdowns in the fourth, but Browns kicker Cody Parkey connected on two short field goals to prevent the comeback attempt.
Mayfield completed 21 of 34 passes for 263 yards and three touchdowns. Chubb had 18 carries for 76 yards, while Hunt recorded 48 yards and two touchdowns on eight rushes.
Steelers signal-caller Ben Roethlisberger was 47 of 68 passing for 501 yards. He threw four touchdowns and four interceptions.