Half-time substitute Ianis Hagi unlocked a stubborn Livingston defence as Rangers progressed to the Premier Sports Cup semi-finals following a 2-0 Ibrox victory.
The Romania international set up Kemar Roofe to break the deadlock with little over two minutes of the second half gone.
Livingston then began to shoot themselves in the foot and Alfredo Morelos scored his first goal in six matches following a gift from visiting goalkeeper Max Stryjek in the 63rd minute.
David Martindale’s side did not manage any efforts on goal whatsoever.
Rangers ensured they would go at least one stage further than they did in both cup competitions last season as Steven Gerrard bids to win his first knockout tournament as manager.
Gerrard made six changes following the weekend draw with Motherwell with Morelos restored to the front line.
Livi handed Stephane Omeonga a debut while Cristian Montano returned for his first appearance since the group stage.