United had only scored 12 times in 16 games in front of their own fans and it needed something special to swing the game in their favour after a dominant start by the visitors.
Ex-Leeds trainee Tom Cairney gave Fulham a deserved 17th-minute lead after finding the net for the fourth time in as many games but Cook ended United's six-hour wait for a goal on 38 minutes to put the hosts on level terms at the break.
Both sides pushed for a winner as the clock ticked down but the game ended in stalemate despite Fulham defender Fernando Amorebieta's stoppage-time red card for a second bookable offence.
Protest group 'Time To Go Massimo' beamed a series of 30ft high images on to the side of Elland Road before and during Leeds' last home game against Middlesbrough but Tuesday's match passed without incident.
The home side survived an early scare as former youth player Luke Garbutt's deflected strike flashed just past the post and Sol Bamba came to the rescue soon after to head ex-United captain Ross McCormack's mishit volley against his own crossbar and to safety.
But United's luck ran out when another old boy Cairney took the ball down on the edge of the box and volleyed past the motionless Marco Silvestri.
Fulham had their tails up and almost straight from the restart Garbutt got in behind the Leeds defence only to see his weak effort comfortably saved by Silvestri.
Bamba blazed over the bar from eight yards and Tom Adeyemi did likewise with a better effort as United began to get a foothold in the game.
Back came Fulham and they twice went close to doubling their lead as Lewie Coyle, on his first league start for Leeds, prevented McCormack from tapping home Ryan Fredericks' dangerous ball in and Garbutt stung the palms of Silvestri.
Leeds were in desperate need of a moment of magic and Cook provided it with a sensational shot which left 'keeper Andy Lonergan with no chance and completely changed the mood inside Elland Road.
Souleymane Doukara then saw a shot saved by the legs of Lonergan and Mirco Antenucci headed straight at the former United goalkeeper as the hosts finished the first half on top.
A well-timed slide challenge by Bamba denied Moussa Dembele the chance to get a shot away after running clear of the Leeds defence and McCormack sent a 30-yard free-kick a couple of yards over the bar in a relatively uneventful opening to the second period.
The visitors were looking the most likely and should have retaken the lead when McCormack found space and squared the ball to Ryan Tunnicliffe, only to see his team-mate fail to connect with his shot before backheeling another effort into the grateful arms of Silvestri.
Cook sparked Leeds into life again with a low shot which went wide and then laid on a great chance for recently-introduced substitute Stuart Dallas, who sent his strike the wrong side of the left-hand post.
Silvestri had to be at his best to keep out a stinging effort from Alex Kacaniklic and Charlie Taylor's 25-yard free-kick was deflected over in a frantic finish.
Amorebieta was sent off for halting a United break with his hand but there was not enough time for the home side to make the one-man advantage count.
Source : PA