The young Gers had the better of the opportunities in a contest played in a good spirits, although they were left to rue the concession of a goal from a corner with 15 minutes to play.
There was a mix of youth and experience in the Rangers ranks, with Sam Lammers leading the line and flanked by Callum Burnside, ten years his junior, and 18-year-old Archie Stevens.
And the Dutch forward was involved in the action early in the first half when an early delivery from Robbie Fraser was on the money, but the 27-year-old could only steer his first-time finish wide of the target.
There was nothing to separate the sides as an energetic Leeds side hustled and harried the Gers, with a powerful shot from distance forcing Lewis Budinauckas into a smart stop down to his right.
Rangers were deriving particular joy down the left-hand side through Fraser, who picked out Josh Gentles on the edge of the penalty area and the 16-year-old forward spooned his finish over the crossbar.
Half-time prompted a triple change for the Light Blues as Lammers was withdrawn for 2007-born Chris Eadie, with Rangers immediately going on the attack through Stevens whose low finish was thwarted by the Leeds goalkeeper.
Kerr Robertson was next to test his luck when the midfielder took aim from 25 yards, but, much to the relief of the visitors, his rasping effort cannoned off the far post and away from the danger area.
A smart pass from Jack Harkness almost engineered an opener for the Light Blues 20 minutes from time, but the angle continued to narrow for Eadie who contrived to steer his finish into the side netting.
Rangers then suffered a setback five minutes later when an outswinging corner was spilled by Alfie Halliwell in the Gers goal and the away side ruthlessly punished the youngster with a finish into the far corner.
Opportunity knocked for Robertson and Eadie before the conclusion of the contest, but Leeds continued to defend their penalty box with vigour and preserved their slender advantage in this afternoon’s pre-season encounter.
RANGERS B: Budinauckas (Halliwell, 60), King (Webster, 60), Allan (Allen, 60), Webster (Harkness, 45), Fraser (Scott, 60), Rice (Robertson, 45), Nsio (Ishaka, 60), Gentles, Stevens, Burnside, Lammers (Eadie, 45)
LEEDS UNITED UNDER-21s: Mahady, Cresswell, Ferguson, Toulson, Lopata-White, Montiero, Vincent, Coleman, Pirie, Chadwick, Richards
Subs: Ombang, Brockie, Moore, Pickles, Render, Dawber