An end to end game featuring a plethora of chances for both sides looked to have been decided in the 77th-minute when Tom Cairney cut inside the area and curled a beauty into the far corner.
But after missing a host of easier chances, Wood stuck away the most difficult one of all when acrobatically tucking away a classic overhead kick in the third minute of injury time.
His effort brought new Whites head coach Garry Monk his first point of the season at the third attempt, ahead of Saturday’s Yorkshire derby at Sheffield Wednesday.
There were very few chances in the opening exchanges and a seventh-minute Fulham corner was well dealt with by United whose Kyle Bartley beat Matt Smith to win a header and clear.
Smith then had half a chance six minutes later when being picked out by the superb Ryan Sessegnon’s left flank cross but the striker headed straight at Rob Green.
A Sone Aluko shot was then blocked by Bartley as Fulham pressure threatened to build before Stuart Dallas fired a cross-cum-shot straight at David Button from the edge of the area.
Kemar Roofe then earned United a corner soon after but a total waste of a set piece allowed Fulham to break and the Cottagers went very close to scoring when Sessegnon’s cross was headed just wide by the dangerous Floyd Ayite on 20 minutes.
At the other end, Dallas threatened to break through in the Fulham area before the ball was cleared but United then squandered a golden chance through Marcus Antonsson in the 28th minute.
A Fulham mix up in midfield gave the striker an invitation to run in on goal and the Swede quickly advanced on ‘keeper David Button but then rolled his effort just wide of the right hand post.
But Leeds were then handed two big reprieves of their own, first when Green appeared to bring down Aluko inside the area and near the byline, only for referee Chris Kavanagh to award a goal kick.
Then, moments later, a lovely Fulham move, in which Ayite featured heavily, ended with Aluko hitting the inside of the post, allowing Bartley to desperately clear.
The same player then fired over moments later from the outside of the box and when Leeds tried to rally, Wood’s scuffed shot sailed harmlessly wide.
But five minutes before the interval a lovely Whites move ended with Antonsson going close when connecting with an inviting Dallas cross which led to the ball spinning across goal and to safety.
Fulham continued to look dangerous on the counter attack but the half ended with Wood putting a decent header off the post and narrowly wide though the striker was then flagged offside.
There were no changes after the break, after which Dallas immediately volleyed over from 20 yards.High, wide and not so handsome.
But Fulham soon began to pile on the pressure and Green was forced to produce a brilliant save to push away a rising goalbound effort by Ayite for a corner.
Fulham went even closer from that corner as Smith’s towering header cannoned back off the post to safety.
United were getting swamped but the Whites rallied and Button needed safe hands to keep out decent efforts from Roofe and Pablo Hernandez in quick succession.
Then, just moments later, Ayite fired just over from the edge of the area for Fulham.W
Garry Monk then withdrew Dallas for Ronaldo Vieira and United had the next chance when a clever through ball from Hernandez found Wood whose effort sailed just over the crossbar from the edge of the area.
Fulham then squandered another fine chance when Ayite broke clear in the United box but whipped his effort well wide.
After Ayite fired another shot wide for Fulham and Vieira fired over for Leeds, Monk brought on Alex Mowatt for Hernandez with 19 minutes left and from a Whites corner soon after, Antonssson fired over at the far post.
But a game that was miraculously still goalless was finally unlocked with 13 minutes left courtesy of a screamer from Tom Cairney.
A powerful Bartley shoulder charge on Cairney gave Fulham a free-kick on the edge of the area and while Aluko’s goalbound strike was blocked, the ball worked its way to Cairney who cut inside from the edge of the area and curled a beauty inside Green’s right hand post.
Monk reacted by making his final change in withdrawing Roofe for Hadi Sacko but there was no immediate response until Antonsson’s cross found Wood who put a free header from three yards out down into the ground and over the bar.
For Wood, and Leeds, it looked like being another one of those days but the striker then salvaged the Whites a point in sensational fashion in the third minute of injury time.
A Charlie Taylor cross was flicked goalwards by Bartley and, with his back to goal, Wood produced an acrobatic overhead kick which bamboozled Button and nestled in the far corner.
Leeds United: Green, Ayling, Cooper, Bartley, Taylor, Phillips, Hernandez (Mowatt 71), Dallas (Vieira 58), Roofe (Sacko 78), Antonsson, Wood.
Subs not used: Turnbull, Coyle, Bamba, Doukara.
Fulham: Button, Odoi, Madl, Kalas, Cairney, Parker (Tunnicliffe 83), McDonald, Sessegnon, Ayite (Jozabed 87), Aluko, Smith (Woodrow 65).
Subs not used: Joronen, Malone, Ream, Christensen.
Referee: Christopher Kavanagh.
Attendance: 21,204
Read more at: http://www.yorkshireeveningpost.co.uk/sport/football/leeds-united/match-report-leeds-united-1-fulham-1-wood-s-late-bicycle-kick-rescues-first-point-for-whites-1-8072632