Tony Mowbray's men were superior throughout, and although denied a host of chances by a stubborn Barnsley defence, Hines' header on the hour mark had proved enough to guarantee all three points.
The Tykes' fourth straight defeat on the road was confirmed when Ogbeche lobbed Luke Steele as he tracked back from a desperate rush toward the opposition's goal deep into stoppage time.
Jason was the first of the two Steeles to be tested as Andy Gray jabbed into his gloves.
Then, Julio Arca twisted and turned on the edge of the visitors' box. After allowing the ball to drift out of his feet, he wrested back control to ping high above Luke Steele's crossbar.
The Argentinian midfielder nearly turned provider in the seventh minute, his early cross from the left missed by the onrushing Marvin Emnes before Lukas Jutkiewicz just reached it on the byline to nod wide.
Craig Davies inspired what little creativity Barnsley showed in the first half. As Boro failed to clear their lines, he danced forward to pump the ball into the hoardings.
The hosts soon answered back, Emnes stretching to poke Barry Robson's cute pass against Luke Steele.
A clear chance went begging as Richard Smallwood directed a sweet through ball to Justin Hoyte on the right side. The ex-Arsenal trainee clipped a delightful ball across the six-yard box, Jutkiewicz's toe arriving just too late.
With Barnsley content to camp out in their penalty area, Nathan Doyle failed to pick up an Emnes run, the Dutchman wriggling free to blast into the side netting.
But Doyle would then enjoy the Tykes' best opportunity in a lacklustre opening period, volleying a wayward clearance just over the bar and into the home fans.
Mowbray made a double substitution at the break. Jutkiewicz was replaced by youngster Curtis Main as Faris Haroun took over from Arca in the middle of the park.
The tactical switch was slow to pay dividends, but the consistently bright Emnes soon made a menace of himself, his speculative strike cut out by a brave Stephen Dawson.
Middlesbrough made their long-awaited breakthrough on the hour mark. Robson directed a regulation corner into the box and there was no volunteer willing to deny Hines his headed conversion.
Haroun would not trouble the scoreline further with a misguided strike from distance just four minutes later, but Boro were now firmly in the ascendancy.
Chris Dagnall was withdrawn for Jay McEveley as Keith Hill ordered his charges to retreat.
McEveley looked set to rescue a point with 89 minutes on the clock, but Jason Steele dropped quickly to smother his goalbound header on the line.
It was Boro's day though, as two minutes into stoppage time Ogbeche caught Luke Steele in the wilderness to lift a delightful lob high and into the Barnsley net.