Having won 4-0 in the reverse fixture in Gibraltar, Rangers' progression into the next round was all but secured going into Friday's (AEST) clash.
And it finished off the tie in style as Morelos helped illustrate the gulf in class with a treble that followed Joe Aribo's third-minute opener.
Jermain Defoe added a brace and Rangers will next face Luxembourg's Progres Niederkorn, which humiliatingly dumped it out in the first round of qualifying two years ago.
Aribo cleverly tucked home from a tight angle to break the deadlock before Jake Hastie had a goal ruled out for offside in the build-up.
Hastie then fired narrowly wide after good work down the left from Morelos, who made it 2-0 in first-half stoppage time when he nodded home after a corner was flicked into his path.
Morelos converted from the penalty spot 10 minutes after the restart when Greg Docherty was brought down and went on to complete his hat-trick with a back-heeled flick.
Defoe had the final say with a late double off the bench as Rangers reached double figures in a one-sided tie.
The substitute slotted home Matt Polster's right-wing cross to make it 5-0 and then applied further gloss to the scoreline when he was slipped through to slide the ball under goalkeeper Jamie Robba.