It is the best run in Reading's 134-year history but they should have had the game won before a mistake allowed Leeds to grab a late equaliser.
In a frantic closing 15 minutes both sides had the chance to snatch a winner.
Reading keeper Marcus Hahnemann made a superb save from Jonathon Douglas.
And as the match moved into injury time, Neil Sullivan dived low to make an instinctive save and keep out Dave Kitson's close-range effort.
The goal that had been so long coming for Reading finally arrived in the 64th minute.
Gary Kelly turned the ball behind for a corner which was taken by American international midfielder Bobby Convey. The ball was cleared back out to Convey and this time he turned it inside to midfielder Brynjar Gunnarsson.
The Icelander calmly tucked the ball low into the corner of Sullivan's net for his third goal of the season.
It was another member of Reading's Icelandic contingent who played a huge part in the Leeds goal with Ivar Ingimarsson's misfortune delighting the Leeds fans.
They had seen the Yorkshire side forced back by a confident Royals who started the game at a high tempo and maintained it throughout.
Glen Little was the first to come close with a right-foot volley over the bar, then Kevin Doyle just failed to get his foot to Graeme Murty's ball across the face of goal.
Leeds' only effort in the opening stages was a Robbie Blake free-kick from long range that was comfortably saved by Hahnemann.
Kitson then set up a chance for Harper but he failed to connect properly and the ball trickled wide of the upright.
That was the pattern of the game throughout the first half with Leeds clearly happy to be in with a chance after surviving the onslaught.
Reading's opener looked to have earned them all three points until that Ingimarsson blunder.