Lewis Richardson scored again but couldn't prevent a third defeat in four games for the Claret in the U18 Professional Development League.
United took the lead through Norwegian winger Niklas Haugland with 30 minutes played.
Richardson headed home the Clarets' equaliser for his sixth goal in just two games.
But the home side regained the lead moments into the second half through an Edon Pruit own-goal.
The Clarets started brightly and came close early on, as Kai Smith nodded Vinnie Steels’ pin-point cross just wide.
However it was the hosts who broke the deadlock as Haugland slammed home Jenkins’ cross at the back post to give United the lead.
Burnley went in search of an equaliser and it came just five minutes before the break.
Fifteen-year-old Richardson glanced home Matty Rain’s free-kick from the right to pull the scores level and continue his prolific streak.
And the visitors came close to taking the lead just moments before the break as Terry Mupariwa curled an effort just over the bar from long range.
However, just 10 minutes into the second half United grabbed the winner as Henri Kumwenda’s cross deflected off Pruti before rolling past a stranded Kai Calderbank-Park.
Despite the Clarets' best efforts at clawing a goal back late on, Leeds' defence held onto the lead to keep the away side at bay.
Burnley: Calderbank-Park, Moss, Pruti, Chima, Rain, Fenlon, Kershaw, Mupariwa, Smith, Steels (George 68), Richardson.