The hosts took a 24-point lead into the halfway stage against their NFC North rivals, but a stirring second-half comeback from Cleveland almost saw the visitors clinch the win in Mike Pettine's first match in charge.
The Steelers took the lead on their first drive as Suisham knocked over a field goal after seeing his side have a touchdown nullified, but their three-point advantage was soon wiped out by a 39-yard kick from Billy Cundiff.
Pittsburgh regained the lead in the closing stages of the first quarter as LeGarrette Blount powered his way into the endzone from seven yards.
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A third score soon followed, and this time it was courtesy of individual play from Le'Veon Bell, who danced between a number of tackles on his way to a 38-yard touchdown.
Suisham increased the difference to 24 points at half time with a late field goal, but Cleveland set about destroying that advantage after the interval.
Undrafted rookie Isaiah Crowell led the way for his side, bursting over from three yards early in the third quarter to give his side a first touchdown of the new campaign.
That was quickly followed by a second as the debutant doubled his personal tally with a lightning 15-yard break through the middle of the Steelers' defence.
There was to be heartbreak for the visitors, however, as Suisham slotted over a 41-yard field goal in the dying stages to clinch a dramatic victory.