In September, the Ibrox side comprehensively defeated Livingston 3-1, St Johnstone 4-0 and Aberdeen 5-0. However, they started the month with a 2-0 reverse to rivals Celtic in Govan.
The latter win over Aberdeen saw Rangers cut the gap at the top to Neil Lennon's side who dropped points at Hibs.
Their form has carried on into this month with a 5-0 trouncing of Hamilton which took Gerrard's men top.
Morelos displayed his quality and importance to Rangers with three goals in four matches, as well as a high standard of performance.
He was instrumental in the wins, showcasing both his prowess in front of goal and selflessness outside the box.
The awards for the Rangers duo follow August's clean sweep for Celtic which saw Neil Lennon and Odsonne Edouard win August's manager and player of the month.