The Clarets took the lead from the penalty spot at the end of the first half when Emiliano Buendia tripped Maxwel Cornet inside the box.
Ashley Barnes, without a goal in 15 months when netting against Fulham at Turf Moor, confidently dispatched the spot kick having sent Emiliano Martinez the wrong way.
The hosts equalised within the first few minutes of the second half when Matty Cash’s foul throw was missed by the officials, and then ignored by VAR, prior to Buendia atoning for his earlier error and steering John McGinn’s cross past Nick Pope.
Mike Jackson’s men could have won it late on when Roberts, who had been foiled by Martinez, squared the ball to Wout Weghorst, but the substitute’s close range effort, with the goal gaping, was blocked spectacularly by Tyrone Mings.
The result saw the Clarets climb out of the bottom three heading into the final game of the season.
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The hosts equalised within the first few minutes of the second half when Matty Cash’s foul throw was missed by the officials, and then ignored by VAR, prior to Buendia atoning for his earlier error and steering John McGinn’s cross past Nick Pope.