United were comfortable for the first eighty minutes of proceedings before Wanderers set up a grandstand finish when Darren Pratley fired home nine minutes from time.
Gary Madine, who proved thoroughly wasteful in front of goal all game, missed a great chance to level when he clipped the post with the goal at his mercy, but Leeds held out to keep their campaign alive as a competitive entity - in front of a huge 6,630 contingent of travelling supporters - the biggest ever at the Macron Stadium.
Doukara’s fourth goal in six matches put Leeds ahead inside the first ten minutes after a howler from Ben Amos and Diagouraga doubled the tally with his first goal since March 2013 six minutes before the interval.
United’s serene progress was checked by Pratley’s close-range effort, but it was the visitors day.
United made three changes, with Diagouraga handed a first full start since joining the club in place of Luke Murphy, while Mirco Antenucci stepped into the breach in attack, with Chris Wood sidelined with a recurrence of his hamstring problem.
United’s other change came in the back four, with Sol Bamba restored to the starting line-up in place of Liam Cooper, out with a hip problem.
Both sets of fans in freezing conditions were treated to an open lively start to proceedings, with Bolton in better mood on the pitch at least after successive home wins against Eastleigh and MK Dons.
It was Wanderers who had the first half-chance, with a tangle of legs between Liam Feeney and Charlie Taylor eventually seeing the Bolton winger emerge with the ball and he teed up home captain Pratley, who blazed into the massed ranks in the away end.
Leeds fired a quick counter with Lewis Cook, handed an attacking brief in the ‘number role’ in a 4-2-3-1 system, seeing his shot held by Amos, who had a moment to forget not too long after to enable United to take the lead on eight minutes.
Slick play on the counter saw Stuart Dallas, lively from the off, slip in Doukara, who showed commendable poise and control before firing a low shot towards goal, which was fluffed badly by Amos, who diverted his standard effort into the net - to provide the striker with a gift-wrapped goal.
It was the perfect tonic for United and their travelling army, although Bolton spurned a heaven-sent chance to level not long after following some weak defending.
Giuseppe Bellusci’s clearance was picked up by Feeney, whose cross deceived Bamba, with Madine having the goal at his mercy in the six-yard box, but he showed a distinct lack of composure to fire over.
It was a let-off for Leeds, with Bamba then blocking an effort from Wellington Silva before the visitors composed themselves and started to piece together some pertinent play on the counter-attack which started to unhinge Wanderers.
Dallas soon fired off target before Madine went down at the other end under pressure from Bamba and Bellusci, but Andre Marriner was unmoved, just as he was from the resulting corner when a half-shout for a handball was dismissed.
With Cook, Doukara and Dallas in particular troubling Bolton with their movement, Leeds soon conjured another good chance following some flowing football on the break.
Fine work from Doukara and Cook laid the platform before the latter played in Antenucci, whose saw his fierce low shot turned away adeptly at his near post by Amos.
From the resulting corner, Bellusci’s cross-shot was not too far away in another sign of United’s confidence, with Bolton looking suspectible and ripe for the taking with a second goal.
Dallas saw an effort fly off target again before Taylor’s free-kick curler drifted wide after Liam Trotter was penalised for a foul on Dallas.
But Leeds pressure eventually with another fine counter yielding a second goal before the break.
Leeds poured forward after more slick play on the break before Antenucci picked out Diagouraga with a lovely pull-back after he steered the ball low past Amos for a classy second goal on 39 minutes.
The half-time whistle saw Bolton recieve a few boos from the home punters who turned out, with Leeds entitled to salivate at the prospect of more goals in the second period if they continued in the same vein as the first half.
Wanderers produced the first chance of the second half and it was an excellent one with Rob Holding’s header falling invitingly in the six-yard box for Madine, but his close-range effort was diverted for a corner by Bamba, who got in a key block.
A home break then saw Mark Davies fail to pick out a home player in the box while at the other end, Leeds got in a couple of very promising positions, but failed to come up with a cogent final ball.
Leeds did eventually piece together a presentable chance to make it three on the hour mark after another stirring break ending with Dallas picking out Scott Wootton, but he showed a distinct lack of composure in front of goal and fired over.
The visitors’ polished raids on the break were less plentiful in the second 45, but Bolton could not conjure much at the other end, with Madine’s miss on restart representing a big moment.
Amos had to be alert to tip over a well-executed free-kick from distance from Bellusci as Leeds sought to seal the deal ahead of Bolton making a double change with Jay Spearing and Kaiyne Woolery replacing Mark Davies and Wellington.
It failed to significantly pep up the hosts, who did manage to piece together a half-chance with a well-struck shot from Feeney flying wide.
A more substantial chance arrived 12 minutes from time at the feet of Pratley, but his goalbound strike inside the box was blocked in terrific fashion by Bellusci.
Bolton needed some inspiration and found some nine minutes from time when the visitors failed to clear following a Woolery corner with Holding’s header latched onto by Pratley, who was left free to hammer the ball home into the roof of the net from close range to give the hosts a lifeline ten minutes from time.
Suddenly, Leeds’ looked vulnerable and Bolton spurned a huge chance to restore parity when Madine’s bad day was compounded when he struck the outside of the post after the ball fell to him invitingly after Marco Silvestri weakly parried Pratley’s shot into his path.
Evans revealed his pride at the win - and reiterated to Premier League clubs that Lewis Cook is not for sale following his stellar performance.
Evans said: “I stand here immensely proud of that performance on the pitch and immensely proud of the performance off the pitch of the fans. The atmosphere from before kick off to the end was simply stunning.
“For eighty one or eighty two minutes on the pitch, we were sensational on the pitch and then you think the magic of the cup may come back and bite you. We were so far in front and we just needed the third goal. If that had come, it would have been many more.
“But we just wanted to be in the draw.
“When people talk to me how much Lewis Cook is worth, people got an example today. It is why when six or seven in the Premier League are offering multi-million pounds offers and they are here today. I tell you what I am doing if I am the chief scout of a Premier League club, I am phoning my chairman and saying ‘buy him.’
“But he is not for sale and we are not going to sell him. We want to be building with gifted young players on the park like him. “