The winning moment belonged to 17-year-old midfielder Findlay Curtis who confidently dispatched his winning spot-kick, with the young Gers also indebted to goalkeeper Mason Munn who made two crucial stops.
It was a tale of two halves for Stevie Smith’s side as they fell a goal behind on the stroke of the interval, before Chris Eadie was fouled inside the penalty area moments into the second period and he stroked his penalty cooly into the bottom corner.
Indeed, it was Rangers who made the brighter side as Curtis went close when he manufactured space for a shot at the far post but dragged his finish inches wide of the bottom corner.
Chris Eadie was next to test his luck when he manufactured space on the edge of the penalty area and unleashed at goal on his left side, although his shot arrowed over the crossbar.
Quality was conspicuous by its absence in the first half, however, as the home side failed to put the finishing touches on a sweeping move when Calum Adamson opened up his body and steered an effort inches over the crossbar from the left-hand side.
For all of Rangers’ endeavour, they found themselves behind going into the interval when Brandon Forbes squeezed home an effort that ricocheted off Mason Munn and trickled over the goalline from a set-piece situation.
However, the young Gers didn’t allow their heads to drop as they emerged for the second half, finding the leveller just moments into the second period when Eadie went down under the attentions of a Dundee United defender.
The 16-year-old assumed the responsibilities and planted a confident finish into the bottom corner to restore parity and give the Gers a much-needed lift.
The second half then belonged to Rangers as Adamson firstly had a deflected effort tipped over the crossbar by the Dundee United goalkeeper at full stretch.
Paul Nsio then nodded a succession of headers wide following a pair of excellent Blaine McClure deliveries that put the visitors on the back foot.
A scramble in the penalty box ensued with 15 minutes remaining that saw Eadie poke an effort towards goal, but United continued to throw bodies in front of Rangers efforts as they looked to keep their cup campaign alive.
A slick move down the right-hand side allowed a delivery to travel all the way to the far post that looked poised for Curtis to nod home, but Keir Bertie found a pivotal glancing header to knock the ball away for a corner.
Substitute Callum Burnside was then inches away with what proved to be the final opportunity of normal time for Rangers when he blazed a shot over from inside the penalty area.
The sound of the referee’s full-time whistle meant penalties would separate the sides, with Archie Stevens nominating himself to take the first spot-kick that the 18-year-old dispatched confidently into the bottom corner.
Rangers were then praising the contributions of Munn as the 17-year-old goalkeeper thwarted Forbes down to his right to give his side the advantage going forward.
Eadie, who had converted from 12 yards in normal time, capitalised on the save by, like Stevens before him, stroking a cool finish into the bottom corner.
Rory McLeod would halve the deficit for Dundee United, but Rangers moved to within a spot-kick of victory when Burnside fired straight down the middle before another smart stop from Munn put his side in poll position.
The deciding moment fell to Josh Gentles and, despite the 16-year-old being denied, it was to matter little as Curtis was the coolest man in Milngavie to send the goalkeeper the wrong way next up and book his side’s place in next month’s final at the national stadium.
RANGERS: Munn, Hutton, Grant, Wyllie, Scott (Campbell, 86), McClure, Adamson (Burnside, 83), Nsio (Gentles, 70), Curtis, Stevens, Eadie
Subs not used: Halliwell, Kerr, Robertson, Willox
DUNDEE UNITED: Haldane, Constable, Bertie (Borland, 50), Cleall-Harding, Petrie, Domeracki, Carnwath (Stirton, 65), Dewar, Cameron, MacLeod, Forbes
Subs not used: Wilson, Trotter, Welsh, Holt, Gilligan
REFEREE: Josh Hay