They did so just in time, their first victory on the road for 11 weeks cutting Manchester United's lead down to three points and keeping Newcastle's Premiership challenge alive.
However, there was also bitterness in victory as Kevin Keegan strongly criticised Alex Ferguson, his Manchester United counterpart. Keegan appeared to suggest that Ferguson had attempted - and apparently succeeded - in stoking up Leeds for last night's game.
After a 10-man Leeds had pushed Manchester United hard at Old Trafford last week Ferguson accused them of "cheating" their manager and complained that they tried harder against Manchester United than anyone else.
Leeds certainly tried hard last night and Keegan appeared to suggest that the reason for that was Ferguson's remarks.
"Some of the things that have been said in the last few days have been almost slanderous," an emotional Keegan said on Sky television after last night's match. "I think you should send a tape of that [match] to Alex Ferguson. That's what he wanted, isn't it?
"You don't say what Alex said about Leeds, you don't say that in football. He's gone down in my estimation. Football in this country is honest."
Keegan also spoke out over suggestions that it was wrong for Newcastle to be playing in the League against Nottingham Forest at the City Ground on Thursday and to return there the following week for Stuart Pearce's testimonial.
There have been suggestions that Forest might not want to beat Newcastle as it might deter their fans swelling Pearce's income. Frank Clark, the Forest manager, had reacted angrily to these rumours at the weekend.
Keegan added: "I would love it if we beat them [Manchester United], love it. The battle is still on, they have not won it yet."
Manchester United are, however, still strong favourites. While Newcastle took three points, they managed only one goal, from Keith Gillespie after 17 minutes.
Manchester United are six goals to the good and, realistically, Newcastle must gain at least a point at Forest to take the title race to the final day.
Had luck, for once, not favoured Newcastle at Elland Road it could have effectively been over already. Twice in the first 10 minutes, while the black and whites were still grey with tension, Leeds struck the woodwork.
The first was inside three minutes, Lucas Radebe hitting the bar from 20 yards after Philippe Albert had let in Brian Deane. Then Andy Gray glanced a header against the post after Wetherall had laid off Gary McAllister's cross.
Then, in the final minute, Leeds nearly snatched a point. Wetherall's close-range header was bound for goal when the big hands of Shaka Hislop intervened and clutched it tight to his breast.
In between Newcastle had a sheaf of chances, but took only one. That followed Newcastle's first period of sustained pressure, a series of corners, the last of which was well won by Les Ferdinand. His header back across the goal came to Gillespie. He alertly diverted it inside the far post, which had just been deserted by Nigel Worthington.
The goal settled Newcastle but, unlike their Old Trafford rivals on Sunday, they were unable to grab a quick second. Peter Beardsley and Ferdinand brought good saves from Mark Beeney but, after that, it was all misses.
Rob Lee was the worst culprit, astonishingly screwing a loose ball wide from eight yards a minute before half-time after Ferdinand had challenged for Gillespie's cross. Seven minutes later Ferdinand was the guilty party, pulling a 20-yard shot wide after Robbie Elliot had given him a clear sight on goal.
Faustino Asprilla, who was withdrawn after a histrionic display, and Albert shot over, Gillespie fired wide and there were a number of scrambles. None went in and Newcastle thus visit the City Ground on Thursday needing goals as well as points. But at least they are still in the hunt.
Leeds United (4-5-1): Beeney; Kelly, Wetherall, Beesley, Worthington; Gray, Ford (Masinga, 84), Radebe, McAllister, Speed (Wallace, 52); Deane. Substitute not used: Tinkler.
Newcastle United (4-4-1-1): Hislop; Watson, Peacock, Albert, Elliot; Gillespie (Barton, 82), Beardsley, Batty, Lee; Asprilla (Clark, 65); Ferdinand. Substitute not used: Srnicek (gk).
Referee: K Burge (Tonypandy).
Race for the title
P W D L F A Pts GD
Man Utd 37 24 7 6 70 35 79 +35
Newcastle 36 24 4 8 64 35 76 +29
Nottingham Forest v NEWCASTLE
NEWCASTLE v Tottenham Hotspur
Middlesbrough v MANCHESTER UNITED