The Steelers took the lead when kicker Shaun Suisham split the uprights in the first quarter, but a second field goal attempt was blocked.
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Suisham's miss allowed the Browns to edge ahead early in the second quarter when rookie running back Isaiah Crowell powered over from five yards out.
A strong performance by the Browns defense ensured that Pittsburgh twice failed to secure a first down on back-to-back drives.
Cleveland extended their advantage further before the end of the second quarter when Hoyer handed off to Tate, who sprinted over for Cleveland's third touchdown of the day.
Kicker Billy Cundiff added a field goal with the only scoring play in the third quarter to put Cleveland 21 points clear heading into the final 15 minutes.
Tate was involved again early in the fourth, carrying the ball to the one-yard line before bursting over for his second touchdown of the day.