Goals from Lisa Martinez and Megan Bell had the Gers 2-0 ahead at the break, with centre-back Martinez netting again in the second 45.
Malky Thomson and Gregory Vignal named three changes from the side which beat Queen’s Park on Thursday night as Jenna Fife, Daina Bourma and Chelsea Cornet came into the starting eleven with Megan Cunningham, Clare Gemmell and Bala Devi dropping to the bench.
Megan Bell was first to have an effort for the Gers as she got on the end of a long-ball forward from Lisa Martinez but her shot was held by the Hamilton goalkeeper.
Rangers were ahead after just 13 minutes as Lisa Martinez got on the end of a Demi Vance corner to turn it behind Sarah Rhind.
Megan Bell doubled the lead on the 22nd minute as she was through on goal with the keeper and calmly slotted beyond Rhind for her fourth goal in three games.
The Northern Ireland international came close again shortly after as she got on the end of a Chelsea Cornet cross but the 18-year-old’s effort was just wide.
Centre-half Martinez netted her second of the game on the 68th minute as she headed home a Megan Bell corner.
Daina Bourma came close to adding a fourth a few minutes later as she got her head on the end of an Emma Brownlie cross but it was saved by Sarah Rhind.
Rangers were controlling the SWPL Cup tie with Chelsea Cornet shooting just over with a looping effort from outside the box.
Daina Bourma had a header saved late on as the Gers won their third game on the bounce.
Rangers: Fife, Brownlie, Martinez (Girasoli, 73), Vance, O’Neill, Reilly (Gemmell, 69), Cornet, Docherty, Bell, Bourma, Hay (Devi, 69). Subs: Cunningham, Mcquillan, Macdonald.