The young side scored their full complement of spot-kicks through Luca Rankin, Coben Fergusson, Ben Hutton and Calum Young, with Sam Reid making a big save between the posts.
In truth, it was a fully merited triumph for the young side who broke the deadlock through a familiar source when Ashton Patton fired home from close-range.
They doubled their advantage after the interval when Ben Hutton fired home from a corner, but Dundee United fought back from a Jack Anderson double to force the decisive penalty shootout.
Rangers' name will now be in the hat for the next round, and attention now moves to midweek as the Under-19s make the trip to Poland to face Dynamo Kyiv in the first leg of their UEFA Youth League second round clash.
With only players born in 2009 or younger eligible for the competition, Stevie Smith named several changes to his side from the KDM Evolution Trophy clash against Peterhead last week.
Among those who retained their place in the starting lineup was Ashton Scally who continues to build his minutes following his return from injury.
It took some time for the first attempt in anger as the ball broke to Ben Hutton on the edge of the penalty area and the midfielder curled off target.
The 17-year-old threatened just moments later when Scally wriggled free in midfield before releasing Coben Fergusson. The winger jinked his way to byline before standing up a tantalising cross in which Hutton volleyed only inches wide.
Rangers were beginning to find some momentum in the contest as Fergusson showed his quick feet on the left-hand side to create a shooting opportunity in which he forced a smart save, albeit at a comfortable height, for the hosts’ goalkeeper.
Played in the shadows of Tannadice, the rain was relentless on Tayside as the Gers looked to take advantage by moving the ball quickly.
Down the opposite end, Sam Reid was forced into a reflex save when Dundee United pieced together a threatening move and Emil Franczak curled dangerously towards goal first-time from the edge of the danger area.
After finding their feet in the contest, the Gers certainly merited the opening goal when it arrived just shy of the 25th minute.
Fergusson was at the heart of the move once again on the left, dancing his way through a sea of tangerine just inside the penalty area before laying the ball beautifully into the path of Patton who continued his fine goalscoring form by guiding his finish into the roof of the net from close-range.
Opportunity then beckoned for Ruaridh Hall when the forward left two Dundee United markers in his wake in a swift transition through the heart of the pitch, but after steadying himself inside the penalty area the Scotland Under-17 international contrived to blast over the top.
Moments later, Luca Rankin stung the palms of the goalkeeper with a free-kick from 20 yards out.
Unfortunately the night ended prematurely for goalscorer Patton as he was forced off through injury and replaced by Jackson Barr on the right-hand side of attack.
Despite Rangers’ superiority in the contest, their advantage remained slender against a Dundee United outfit pushing for an equaliser.
Opportunities were at a premium for either side as the minutes continued to eat into the second half, with the combination of Fergusson and Hall, teammates recently at national team level, looking lively.
On his first appearance in the competition, substitute Barr went close to doubling his side's advantage when he met a Fergusson knockdown from a Hall delivery but could only glance his header into the grateful arms of the goalkeeper.
The woodwork was rattling just minutes later when Rankin set his sights 30 yards from goal and unleashed a thunderous effort on his left side which cannoned back off the post.
From the subsequent corner, however, the young Gers eventually found the all-important second goal.
The delivery was deep towards the back post, but with the aerial presence of Conor Owen rising, the ball ricocheted back across goal for Hutton who had the simplest of tasks in stabbing home at the far post.
Any comfort for the Gers was short-lived as United restored optimism when they pulled a goal back through Jack Anderson who squirmed a finish under Sam Reid at his near post.
He then struck ten minutes later for the home side, flicking into the bottom corner from a low delivery to level the contest and force a penalty shootout.
But the tone for the spot-kicks was set by Sam Reid who flew to his right to deny the first taker, with Rankin and Fergusson converting to pile on the pressure.
Buchanan then struck the post for Dundee United, with Hutton's strike setting the stage for Young to rifle into the bottom corner and deliver the triumph for the Under-18s.
RANGERS: S. Reid, J. Reid, Owen, Ngwenya, Scally (Ramage, 60), Young, Hutton, Rankin, Patton (Barr, 43), Fergusson, Hall
Subs not used: Millarvie, Johnston, McGhee, Molloy
DUNDEE UNITED: Zerebecki, Buchanan, Beverley, Reynolds, Lee, McHale, Leiper, Forrest, Anderson, Connelly, Franczak
Subs: Carrigan, Awa, Popham, Ayeni, Mathers, Ugbethe, Mess
