Leeds showed five changes from Saturday's 2-1 win against Exeter with Andrew Hughes in for injured left back Ben Parker and a new-look back four also including Paul Huntington and Lubo Michalik, in at centre back for Rui Marques and Patrick Kisnorbo, who were away on international duty.
Striker Showunmi, replacing the injured Jermaine Beckford, pounced ater 54 minutes, reacting quicker than goalkeeper David Knight to poke home a teasing cross from Robert Snodgrass.
The Nigerian international had not scored since suffering a major health scare last December when he underwent treatment for a blood clot on his lung.
United also hit the woodwork through Bradley Johnson after some fine footwork from man-of-the-match Mike Grella, but Darlington's young striker David Dowson saw an effort ruled out for offside and then tested Leeds goalkeeper Shane Higgs with a low shot.
Jonny Howson was inches away from a goal with a curling shot that had the keeper beaten and Grella went close with a couple of strikes from the edge of the box.
But Showunmi should have sealed victory 10 minutes after his goal, only to horribly miskick with the goal at his mercy after good build-up work by skipper Howson and Grella.
Leeds could have done with the prolific Beckford as they struggled to break through a gutsy Darlington rearguard in the opening 45 minutes.
Though Snodgrass flashed a volley well wide after just five minutes, Knight was not tested until the 27th minute.
The former Middlesbrough keeper produced a fine double save to block Michael Doyle's superb 20-yard drive and Showunmi's follow-up.
Darlington responded with two dangerous raids. Chris Lumsdon's free-kick was met by Lee Thorpe whose header flew wide, then Thorpe set up Jamie Chandler but he could only fire straight at full-back Jason Crowe.
Luciano Becchio replaced Showunmi on on 73 minutes and Neil Kilkenny made his season bow, replacing the tiring Grella on 83 minutes.
Darlington almost equalised in stoppage-time when defender Mark Bower flashed a header narrowly wide when totally unmarked from a Gary Smith corner.
Forty-year-old Dean Windass came on but could not inspire a Quakers' comeback as Showunmi's first goal in eight months settled the tie as Leeds just about deserved the victory having dictated much of the play.
Darlington: Knight, Arnison, Bower, Foster (Plummer 76), Miller, Gary Smith, Lumsdon, Jeff Smith (Windass 58), Chandler, Thorpe (Main 79), Dowson.
Subs Not Used: Liversedge, Bennett, Riley, Convery. Booked: Chandler.
Shane Higgs: The United goalkeeper had even less to do than he did against Exeter on Saturday. His handling was steady, though, and kept an unfamiliar defence on an even keel - 6/10
Jason Crowe: Darlington were never able to put him under great pressure and the right side of United's defence held firm. Has more to offer in attack than he was able to last night - 6/10
Lubomir Michalik: Helped restrict Darlington to one shot on target, and although Todd's forwards gave Michalik a physical work-out, they had few opportunities to get in behind him - 6/10
Paul Huntington: Made an important block from Chandler 10 minutes before the break and was kept very honest by Thorpe. A 90-minute competitive outing should do him good - 6/10
Andrew Hughes: Picked in preference to Aidy White and did not let Grayson down. Darlington made limited use of the flanks, but Hughes rarely allowed the door to the right wing to open - 6/10
Robert Snodgrass: Giving Snodgrass freedom to roam in the second half was tactically astute and his cross for the goal epitomised his effort to force something from an uninspired game - 7/10
Michael Doyle: Doyle's passing was relatively tidy and the loan man protected Grayson's backline well. Produced the only effort worth mentioning in the first half, an awkward volley from long range - 6/10
Jonathan Howson: The skipper failed to make much impression on the first half but came to life in the second, benefiting from the space and possession he started to see in midfield - 6/10
Bradley Johnson: Everything United did went through him in the first 45 minutes, but his end product was often lacking. Unlucky to see a shot rattle the crossbar, though the miss was not crucial - 6/10
Enoch Showunmi: His anticipation of Snodgrass' cross allowed him to claim the only goal, the highlight of a mixed night. Missed another great chance, and parts of the game passed him by - 6/10
Mike Grella: One of the stars of a mediocre show in Darlington. Dynamic and inventive on the ball, always looking for space and a striker who is growing in confidence by the game - 7/10
Luciano Becchio (for Showunmi, 73) - 5/10
Neil Kilkenny (for Grella, 83) - 5/10
Andy Robinson (for Snodgrass, 90) - 5/10
Subs Unused: Casper Ankergren, Alan Sheehan, Aidy White.