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The Clarets came close to taking the lead on 21 minutes courtesy of Michael Kightly, who managed meet the ball with his head, but his effort came back off the bar.
It wasn't all the hosts' way, though, with Danny Graham and Jacob Butterfield forcing a couple of fine saves out of Tom Heaton shortly before the interval, ensuring the scoreline remained level at the break.
Four minutes into the second half Burnley had a glorious opportunity to take the lead. First, Kightly saw his low cross well defended by Nathaniel Chalobah, before Scott Arfield's effort was cleared off the line.
Just two minutes later Middlesbrough took the lead through Buttefield, who found the back of the net with an assured finish.
Burnley turned up the pressure to force an equaliser, and Ings came close on the hour-mark when he charged through, only to drag his strike wide of the target.
The final 10 minutes saw Burnley attack Boro repeatedly, although a series of saves and near misses ensured the promotion party would have to wait with the away side holding onto their 1-0 lead.