The on-loan Arsenal forward opened the scoring in the 84th minute and earned the penalty - calmly converted by Mateusz Klich - for the second as Leeds won back-to-back games at Barnsley for the first time in their history.
The Reds lost their last home meeting with Leeds 2-0 in October 2017 and a repeat of that scoreline has extended their wait for a second league win of the season.
Barnsley last won on the opening day of the campaign when they beat Fulham at Oakwell to get their campaign off to the perfect start.
However, they have struggled for points since with defeat to Leeds leaving them on five points from their first seven games.
Daniel Stendel will take plenty of encouragement from the Reds display as, despite having less of the ball, they caused Leeds plenty of problems.
United, owing to Swansea's defeat to Nottingham Forest, have gone back to the top of Championship table on goal difference.
Many inside Oakwell were left wondering how they hadn't seen a goal before Nketiah popped up with the decisive opener.
Only 23 miles separate Barnsley and Leeds and the Yorkshire foes served up an entertaining derby but despite goals coming at a premium, the chances didn't.
Marcelo Bielsa described his team's wastefulness in front of goal as "characteristic" in the build up to the game and it was something that almost cost them again.
The visitors created the better chances at an overcast Oakwell but the home side could so easily have won the game themselves.
Both sides looked to get on the front foot with Barnsley constantly harassing the Leeds men in possession.
The visitors enjoyed 66 per cent possession in the first half and should have gone into the interval with the lead.
Pablo Hernandez fired over from inside the area after a cut back from Jack Harrison.
Harrison then had the best chance of the half when the ball came to him at the back post from a corner but his shot smacked the outside of the upright.
Patrick Bamford twice found himself with a free run into the box but was thwarted by Brad Collins who stood up well to keep it goalless.
Barnsley's high press caused the visitors problems and former United man Mallik Wilks stung the palms of Kiko Casilla early on before turning Ben White and firing wide across goal late in the first half.
Alex Mowatt threaded a lovely ball through to Luke Thomas but his shot was kept out by the left leg of Casilla.
Leeds' full-backs, Stuart Dallas and Ezgjan Alioski, pushed high up the pitch when the away side were in possession, leaving them vulnerable when Barnsley managed to win the ball back.
The Reds gave as good as they got and almost went in front early in the second period. Jacob Brown's curling effort was tipped over by Casilla before Wilks pounced on a mistake from White but hammered his shot into the side netting.
Bamford had a goal ruled out for offside while Barnsley continued to hit Leeds on the break but couldn't find the decisive goal before Nketiah's crucial strike helped secure the points for the away side.
Barnsley: Collins; Williams (Pinillos 71), Halme, Sibbick, Anderson; Cavare, Mowatt, Thomas (Thiam 67), Brown (Bahre 79); Wilks, Woodrow. Unused subs: Walton, Chaplin, Green, Oduor.
Leeds United: Casilla; Alioski, Cooper, White, Dallas; Phillips; Shackleton, Hernandez, Klich (Berardi 90+2), Harrison (Costa 45); Bamford (Nketiah 70). Unused subs: Meslier, Douglas, Gotts, Roberts.
Referee: Keith Stroud.