Neil Warnock's men were the better team in the first half, creating a number of opportunities to open the scoring.
Leeds were then denied by another goal-line clearance just moments after the interval, as this time Morison saw his header kept out.
The hosts began to threaten as the second half wore on, and Paddy Kenny was called upon to make a smart double save.
However, he could not keep a sheet clean as he was beaten by Curtis Main's looping header with eight minutes remaining.
Boro finished the match with 10 men as the goalscorer received his marching orders, but they were able to hold out for all three points.