The Clarets looked on course to drop out of the automatic promotion places after Jordan Rhodes put the league leaders in front, but Keane salvaged a 1-1 draw for his side to keep them just two points off the top.
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The visitors came within inches of taking the lead in style after just 13 minutes when Grant Leadbitter's corner was only cleared as far as Albert Adomah, who fired a vicious strike against the crossbar from outside the area.
Boro threatened again five minutes later when Stewart Downing sent a dangerous ball into the box, but it was just too far in front of Rhodes for the striker to turn it in.
Burnley responded with a couple of chances in quick succession as first Joey Barton sent a 25-yard effort bouncing narrowly wide before the newly-crowned Championship player of the year Andre Gray attempted an overhead kick that also went off target.
It was the hosts who were beginning to look the most likely to break the deadlock, though, and they came close again just before the half-hour mark when Gray slipped a pass through to Scott Arfield inside the area, only for the midfielder to fire over the crossbar.
George Boyd then sent a volley high and wide before Boro created the last chance of the first half when Adomah's cross fell to Rhodes, who also couldn't hit the target with a volley from a tight angle.
Both sides came out quickly in the second half and had sights of goal in the opening minutes, but Sam Vokes's shot on the turn trickled harmlessly wide before Adomah could only find the side-netting at the other end.
Burnley began to gain an element of control around the hour mark and came close again when Barton nodded just over, but it was Middlesbrough who finally broke the deadlock with just over 20 minutes of the match remaining.
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Downing whipped a free kick into the box for Daniel Ayala, who headed it across goal where Rhodes was waiting to apply the finish for his third goal in his last two games.
The hosts went in search of an equaliser that would keep them in the automatic promotion places, and they very nearly got one 10 minutes after falling behind when Boyd's effort from inside the box was blocked on the line.
Matty Taylor was the next to threaten when his deflected free kick rolled agonisingly past the post for a corner, but Sean Dyche's side were given six minutes of added time to come up with an equaliser and duly rescued a point late on.
It was an untidy goal as Taylor's corner bounced around in the penalty area before hitting Keane and going over the line as Burnley held on to their unbeaten league record in 2016 by the skin of their teeth.
It is now 20 Championship matches without defeat for the Clarets, who also halted Boro's six-match winning streak to keep themselves just ahead of Brighton & Hove Albion on goal difference following the Seagulls' 4-0 win over Queens Park Rangers.