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Less than 10 minutes later, Smith nodded the ball into the path of Ross McCormack, who scooped the ball goalwards, and it squirmed under the body Steele and rolled into the back of the net.
Barnsley struggled to create many chances as Tomasz Cywka had the home side's best effort, but it was a poor strike from 25 yards.
The second period began in scrappy fashion, with neither side able to build any meaningful attacks, although Leeds did have a shout for a penalty waved away as they claimed for handball.
McCormack showed the kind of form that has seen him net 28 goals in the league this season and flashed a shot over the bar before teeing up Smith, whose shot was blocked by Martin Cranie.
Barnsley pushed forward in the final 10 minutes, but they were unable to get a goal and are now three points from safety with three games remaining.