Steven Naismith opened the scoring at Fir Park, before Kyle Lafferty doubled the advantage and Gregg Wylde wrapped up the victory late on.
The 3-0 win will no doubt act as a much-needed confidence-booster ahead of Thursday’s second leg against the Slovenians at Ibrox as Rangers attempt to overturn a 2-1 deficit.
Rangers handed first SPL starts to youngsters Jordan McMillan and Ross Perry. Wylde and Lafferty were also drafted into the side, with Maurice Edu suspended, Lee McCulloch suffering a recurrence of a knee problem and Kirk Broadfoot also absent from the squad.
Carlos Bocanegra – who made his Rangers debut against Maribor in Slovenia – was unavailable for selection as he and fellow American Alejandro Bedoya failed to obtain visas in time for Sunday’s game.
Rangers had hoped Steven Whittaker would recover from a broken toe but he remained on the sidelines along with skipper David Weir, Kyle Bartley and Sasa Papac.
Motherwell headed into the game in better shape, with Keith Lasley returning from suspension to replace Ross Forbes the only change from their win over St Mirren.
Rangers were under pressure in the opening minute when Tom Hateley whipped a corner into the box for Jamie Murphy in front of goal but Allan McGregor managed to block before he could connect cleanly.
At the other end, a Steven Davis free-kick was met by Nikica Jelavic but the striker nodded just over from 10 yards out, before directing another header wide of target moments later from a Wylde corner.
In an action packed opening to the game, McGregor was called into action to prevent a stinging effort from Murphy finding the back of the net, before Randolph was forced to react quickly and block from Lafferty.
The breakthrough came after 19 minutes when Davis sent a corner into the packed goalmouth and Jelavic was first to react with a bullet header that was helped into the back of the net by Naismith with the final touch.
Motherwell – who had not conceded a goal this season ahead of the visit of Rangers – tried to haul themselves back into the game.
Murphy had the ball in the back of the net after 37 minutes but the effort was disallowed after the Well player was deemed to have handled the ball before unleashing the shot.
Rangers were celebrating again on the stroke of half-time when Lafferty raced on to a pass from Juan Manuel Ortiz before sending an angled drive past Randolph.
Lafferty could have added to his tally after the restart when he again combined well with Ortiz only to see his effort collected by Randolph on this occasion.
The Spaniard also played the part of provider for Naismith but he was wasteful with the shot that was high and wide and did little to trouble the Motherwell goalkeeper.
There was a close call for Rangers when Chris Humphrey’s effort skimmed the crossbar, before Davis saw his own shot crash off the woodwork at the other end of the pitch.
Rangers were threatening again when Jelavic found Ortiz at the back post and he nodded back across goal for Naismith, who saw his own header dip just over the crossbar.
Lafferty, who had made his first start after recovering from a hernia operation, was replaced by John Fleck for the final 15 minutes.
Rangers wrapped up the victory with six minutes remaining when Davis slotted into the path of Wylde and he rifled past Randolph to ensure no way back for the home side.