The Colombian struck in the second half to give Steven Gerrard's men a lead for the return tie in Armenia, despite John Lundstram seeing red shortly before the break.
Save for a Ryan Kent effort that hit the crossbar, Rangers struggled to turn possessional dominance into chances in the opening period.
Lundstram's red for two sloppy yellow cards left the hosts a man down and Calvin Bassey's claims for what would have been a soft penalty summed up a shoddy first-half effort from Rangers.
Scott Wright also felt he was felled in the area by Rumyan Hovsepyan early in the second half before Morelos saw an effort cleared off the line by Didier Kadio and then headed woefully wide from Ianis Hagi's teasing delivery.
But it was third time a charm for Morelos, who drilled under the legs of Ognjen Cancarevic after beating the offside trap to reach James Tavernier's long pass in the 67th minute.
That proved enough for a Rangers side that were knocked out in the Champions League qualifiers by Malmo.