A turgid affair saw the Whites toil against a defence that simply refused to shift, Marcelo Bielsa's men struggling to break down the resistance of the Royals until Jack Harrison's 87th minute header.
It will not live long in the memory as a performance or a contest, but all is well that ends well and that winner, United's second late one in as many games sent Leeds top of the Championship and brought a fourth consecutive victory.
Ultimately, patience and the physical fitness of the team, two staples of the Bielsa regime, came to the fore in a big way once again.
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If the late goal and the 1-0 scoreline maintained the morale around Elland Road, it will have had the opposite effect elsewhere in the division, hopes of a United slip-up dashed once again when a poor result appeared imminent.
That they could not find a way through, for so long, against Reading mattered not one jot to the hordes of Whites behind Reading's goal when the ball left Harrison's head and the net bulged.
Games are played over 90-plus minutes, but they can be won in a matter of seconds, in individual moments that stay with a fanbase for years to come.
The drama that arrived in the final stages was a world away from the fare served up by both sides throughout, but Leeds at least showed positive attacking intent, their hosts bereft of it more often than not.
Reading’s best early moments came from poor moments for Leeds.
The Whites’ inability to make anything of corners has become a sore subject of late, so many have they wasted, and their first at the Madejski Stadium was almost painful – Reading winning the header and racing away on the break, Yakou Meite’s cross just too far ahead of the sliding George Puscas.
Their second corner very nearly brought a similar result, Stuart Dallas having to nip in and halt Ovie Ejaria as the Royals again won the header and countered.
The opening 20 minutes saw a few too many placatory hands raised from Leeds players, passes going astray all over the pitch, stopping the visitors from building any real momentum.
Even Pablo Hernandez and Kalvin Phillips, men whose pinpoint passing accuracy has become a reliable and key aspect of Leeds’ play under Bielsa, were guilty of rare wastefulness in possession.
The Whites dominated the ball but did little with it, their play grinding to a halt the closer they got to goal, allowing Reading to form an impenetrable blue and white wall.
Frustration grew as the half went on and worsened considerably when Tyler Roberts went down with yet another injury, the attacking midfielder summing up the feelings of the Leeds fanbase as he shook his head in disbelief.
His evening came to an end after just 35 minutes, Costa coming on in his place.
Winning corners was about all Leeds could muster in the first half, each of them bringing the same result.
Reading, for their part, did very little to liven proceedings, the dribbling prowess and skill of Ovie Ejaria giving home fans a couple of nice moments but not presenting a real problem for the visitors.
But at least when the Royals won a corner late in the half they made something of it, Meite’s downward header forcing Kiko Casilla to save on his line.
Leeds were shot shy and cross shy, attempting and failing to walk the ball into the net time and time again.
But there was, mercifully, some more intensity to Leeds’ play in the early moments of the second half and they were finally able to cut through the home defence with a couple of quick, incisive passes that both looked for Costa, yet Reading recovered to smother the chances.
It was a little brighter from Leeds, Ben White, evidently feeling his side had things in control from a defensive point of view, stepping forward to offer himself as an attacking option and winning a corner that was, at long last, met by a Leeds player, albeit headed well wide by Liam Cooper.
The Royals continued to present an organised, resolute defensive shell and it took over an hour but Leeds eventually tried to bypass it with a long range shot, Stuart Dallas whacking one past Rafael Cabral and leaving the crossbar shaking.
Reading did get to play with the ball in the second half too, just not very often, yet they did forge a good chance of their own when John Swift’s free-kick fell to Meite in the box, he was prevented from shooting and Michael Morrison took over, blasting high over Casilla’s net.
Dallas went on a one-man shooting spree, aided and abetted by Gjanni Alioski, who had come on for Mateusz Klich, volleying the substitute’s first cross straight at Cabral and his second into the side netting from a tight angle.
As time ticked into the final five minutes there was little to suggest either side would produce anything special, before a chance arrived at either end, the second of which decided the game.
Reading won a free-kick 35 yards from goal and as the rain fell and made the ball and the turf slippy, Swift’s dipping effort could have caused real issues for Casilla, but he was equal to it, beating it away and allowing Leeds to break to devastating effect.
Alioski fed Costa, he curled in a cross that made it all the way to the back post where Harrison arrived to crash home a header and send 2,050 away fans into raptures.
What followed was a thoroughly professional and composed display of game management, Leeds holding on tightly to what they had and leaving the pitch all smiles, on a night that could have ended in grimaces.