For the second Championship game in succession, Leeds looked set to travel back to Yorkshire with just a point but three days after an own goal 90th-minute winner at Luton Town, United struck late to claim all three points.
From a rapid counter attack that began with a Reading free-kick, Whites substitute Gjanni Alioski fed Helder Costa whose deflected right-flank cross picked out Harrison to head home his fourth goal of the campaign.
In netting his third goal in United's last four fixtures, Manchester City loanee Harrison's header sent Leeds top of the Championship and one point above West Brom who host seventh-placed Bristol City on Wednesday night.
"I think it really showed the fitness of our team late on in a game," said Harrison.
"You have all these players making runs forward, big, long, hard runs forward just to try to get that goal in the last few minutes.
"It was a brilliant ball by Helder and I was able to get there on the back post."
Asked if he was conscious of the clock ticking down as United again looked to be heading for a draw, Harrison said: "A little bit. There's a glance up at the clock a few times towards the end of a match.
"Even when the ball goes out for a goal kick or something like that, I always try and quickly get the ball because even if we are drawing I feel like as a team we should be pushing for the three points.
"With the fitness we have, we are able to do that and it showed in the last few minutes.
"We were able to push even more and make those intense runs to try and get the last goal or the winning goal."