Marcelo Bielsa made one change for the second trip south in four days as club captain Liam Cooper replaced Gaetano Berardi in defence.
It was the Royals who started the better of the two sides and they had a golden opportunity to open the scoring inside the first 10 minutes.
Yakou Meite broke down the left and squared a low cross to George Puscas but the striker failed to get his foot on the ball as he lunged inside the box.
United were riddled with mistakes and misplaced passes as the half continued with Patrick Bamford and Stuart Dallas both seeing shots blocked.
Leeds were hit by a blow just after the half hour mark as Tyler Roberts was forced into an early substitution as his injury woes continued with Helder Costa stepping off the bench in his place.
Mark Bowen's men had the best chance as injury-time loomed ahead of the break with Meite jumping highest to fire a header on target from a John Swift corner but Kiko Casilla was equal to it.
The second period continued in a similar pattern to the first with Leeds this time enjoying more of the ball after the interval.
Cooper and Bamford both fired efforts over the goal before Dallas struck the crossbar as he swung an effort from distance.
Reading went close themselves as they lifted a shot over Casilla's net following a panicked scramble in the box but the hosts were dropping deeper as the game progressed.
Leeds, though, left it late for a second game in succession as Bielsa's side broke the deadlock with just three minutes remaining.
Substitute Gjanni Alioski released Costa down the right as United broke away from a Royals corner.
The winger's cross was met by the head of Harrison as he timed his run to perfection to power the ball home at the back post and spark wild celebrations from the 2,000 travelling supporters.
With the victory United hit the summit of the Championship with West Brom entertaining Bristol City on Wednesday night.
Reading: Rafael, Gunter, Miazga, Morrison, Moore (c), Richards, Rinomhota, Ejaria, Swift, Méïté, Pușcaș (Boye 70). Subs: Walker, McCleary, Loader, Howe, Blackett, Pelé
Leeds United: Casilla, Ayling, White, Cooper, Dallas, Phillips, Klich (Alioski 66), Hernandez, Harrison, Roberts (Costa 34), Bamford. Subs: Meslier, Douglas, Berardi, Shackleton, Clarke
Referee: Darren Bond
Attendance: 16,918 (2,050)