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Middlesbrough continued to have the better of the chances and on the half-hour mark Lee Tomlin saw his shot saved by Steele, before Reach smashed wide from the rebound.
Against the run of play the visitors took the lead in the 53rd minute when Jean-Yves Mvoto fired home.
The hosts responded well and equalised a minute later through Tomlin, who latched onto a loose ball outside the box and drilled past Steele.
Graham was close to adding a second goal just moments later, but he saw his free kick saved by the keeper.
In the final minute Graham put the hosts ahead when he slotted in after Albert Adomah's shot was saved, before Tomlin added a third goal in stoppage time to seal Barnsley's fate.