Returning midfielder Ryan Jack scored a brace of strikes to have the Light Blues on easy street by the interval.
Morelos grabbed his second with a drive in the 72nd minute to make it 4-0 and keep Steven Gerrard's side behind leaders Celtic on goal difference.
Important games keep coming for the Ibrox men and they will go into their Betfred Cup semi-final against Hearts at Hampden Park on Sunday with further confidence as this was another test of the squad which was passed with merit.
After a hard-fought win over Motherwell at the weekend Gerrard made five changes, with Wes Foderingham replacing injured Allan McGregor in goal, while Glen Kamara, Jack, Ojo and Morelos all came back in.
In the fourth minute Ojo clipped the post with a right-footed shot after being set up by Joe Aribo on a swift counter-attack.
Then, 10 minutes later, midfielder Scott Arfield played a one-two with Morelos and fired in a right-footed drive which was pushed round the post by Staggies keeper Ross Laidlaw for a corner which came to nothing.
The opening goal came at the end of a slick move involving Aribo and Ojo, who slid the ball across goal for Morelos to steer in at the far post.
The Ibrox side grew in confidence and their second goal came in the 29th minute when Jack pounced on defender Liam Fontaine's clearance at the edge of the box and curled the ball past Laidlaw and into the corner.
County had recovered from 2-0 down to draw 2-2 with Hibernian at Easter Road at the weekend but this was a different mountain to climb for the Highlanders.
Jack's second goal was with a similarly well-place drive after good work by Aribo and Arfield and the game was effectively over by the break.
Aribo rattled the post from eight yards as Rangers started the second half with as much purpose as they had finished the first.
Jack then fired a left-footed effort from 20 yards just wide as the Ibrox men kept the pressure on.
In a rare Ross County attack in the 68th minute, Ross Draper's deflected drive from distance took a deflection and wrong-footed Foderingham but rebounded off the post.
Rangers ran up the park moments later and Morelos hammered in a fourth past Laidlaw from 10 yards before immediately making way for Jermaine Defoe, with the visitors keeping a tight grip on the game with impressive energy until the end.