And they snatched the winner three minutes from time when Harrison nodded home Helder Costa's deflected cross.
A second late winner in four days earned Marcelo Bielsa's men a fourth straight win, with three of them coming by one goal.
And it left Reading four points above the bottom three after a second straight defeat.
Leeds began positively, with six-goal top scorer Patrick Bamford off target with a clever volleyed flick in the opening minute.
Reading forced the first real chance when Meite broke swiftly after a Leeds corner had been cleared and spotted George Puscas running into the area but his low cross was narrowly missed by the Romania forward.
Leeds gradually gained control, mostly through the midfield promptings of Dallas and Mateusz Klich.
But Bamford was struggling to find space up front and was largely kept quiet by the solid home rearguard.
Reading relied heavily on the counter-attack, with Ovie Ejaria doing well to trick his way to the byline.
However, Dallas was alert to the danger, adeptly cut out Ejaria's eventual cross and cleared it to safety.
The clearest chance of the first half fell to Meite shortly before the break, when he headed powerfully goalwards from a John Swift corner, but Leeds goalkeeper Kiko Casilla was well positioned and gathered the ball right on the goalline.
Leeds improved after the break and were soon applying constant pressure.
But it was Reading who created the first opportunity. Keeper Rafael Cabral's huge punt forward was collected by Meite but his final shot was poor and never troubled Casilla.
As Leeds' momentum grew, Dallas saw his fierce long-range effort cannon off the bar with Rafael motionless.
Dallas then volleyed straight into Rafael's arms from an Ezgjan Alioski cross.
But Leeds were not to be denied, with Harrison nodding in Costa's far-post centre after a fast break.