After leading two matches against Newcastle and Manly by 18 points only to lose, they have conceded 97 points in their past two matches.
The Aucklanders gave themselves the slightest hope of revival with second-gamer Carlos Tuimavave's 38th-minute try, but then returned to their tackle-shy ways after the break.
Centre Kane Linnett scythed his way through the defence in the 44th minute, Morgan laid the foundations for second rower Gavin Cooper to cross in the 47th and Morgan's pass put centre Brent Tate over in the 52nd.
Even at 40-6 up with two reserves on the bench, the Cowboys weren't finished shooting.
Hooker Aaron Payne scored from close range in the 62nd minute and then Ashley Graham completed his hat-trick in the 68th.
The Warriors lost lock Elijah Taylor before kick-off when he returned to New Zealand following the death of his father.
They managed a consolation try, winger Manu Vatuvei's 100th of his career, in the final minute.
The contest was as good as over when North Queensland scored four tries in the opening 19 minutes to lead 22-0.
While the performance was worthy of a top four side, wins at the weekend to Melbourne and Manly prevented the Cowboys returning to that company.
Warrior Kevin Locke left the field with a serious ankle injury during the first half..
Warriors coach Brian McClennan all but admitted he was now under pressure to retain his job.
"I'm the bloke steering the ship, steering the bus,'' said the former New Zealand and Leeds boss.
"The pressure comes on me. I accept that.''
Asked what result he expected from that pressure McClennan answered: "I'm not sure as yet. That will unfold I guess.
"All I can say is I'm doing the best I can do, I'm doing what I know and what's worked for us in the past.
"I'm big enough and ugly enough to cop it.''
His North Queensland rival, Neil Henry, said that no matter how well the Cowboys played, they were at the mercy of other results when it came to regaining their top four spot.
"We can only control our games and Manly were pretty good against Souths,'' he said.
"It made this game more crucial, I suppose, with the backlog of teams around us.''
The Cowboys take on St George Illawarra in Wollongong on Monday week.
Henry confirmed injuries to James Tamou (ankle) and Ray Thompson (shoulder) were not serious.