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Both teams scored two tries but the visitors snapped their four-game losing streak courtesy of Jordan Kahu's four-from-four goalkicking and their superior ball control.
Queensland playmaker Morgan was relatively quiet until his team fell behind early in the second half, setting up a try for Justin O'Neill and a Kahu penalty goal before snapping the field goal that effectively sealed the result.
Morgan also kicked in open play for a massive 470m and gave the Warriors back three few chances to counter.
Kicking was generally an area of the game where they outclassed the home side, who are badly missing injured halfback Blake Green.
The result leaves both teams with a 2-4 record.
The Warriors created more chances but made more mistakes, crashing to their first loss at home this season following big wins over the Bulldogs and Titans.
Cowboys coach Paul Green said victory was built on a grittiness that he began to see glimpses of in last week's narrow loss to Melbourne.
"I'm not prepared to say it's turned our season around but it's probably a good start," Green said.
"There's been a shift in how we've prepared. Last week was a step in the right direction but it's important that we've built on that. We showed a fair bit of toughness in defence."
Josh McGuire rivalled Morgan as the game's best player, the tough Queensland second-rower running for a game-high 214m and making a team-high 39 tackles.
Warriors coach Stephen Kearney said his team's lack of clinical finishing hurt them, as it did in last week's late loss to South Sydney.
"One thing that stood out tonight was the execution," he said.
"We made six line breaks to their one or something like that. We had close to double the tackle breaks but if you're not executing the final pass, that sort of stuff builds up. We frustrated our way into making it worse."
Warriors winger Ken Maumalo dotted down in the left corner in the 20th minute but the hosts were trailing soon after when Coen Hess took advantage of a Jake Clifford break.
The Warriors spurned several chances either side of the big backrower's try.
Roger Tuivasa-Sheck had his pass intercepted following a sizzling 50m break and David Fusitu'a's apparent try was ruled out when Adam Blair was penalised for obstruction.
Kahu's penalty on the stroke of halftime put the visitors 8-4 up before Tuivasa-Sheck lit up Mt Smart Stadium with a trademark solo try, prising open the defence with his lethal left-foot step.
One jink took him between Clifford and Hess before another left fullback Te Maire Martin grasping at thin air.
However the lead was short-lived, with Tuivasa-Sheck's error gifting the Cowboys an attacking scrum. Morgan's grubber on the first tackle set up former Warriors winger O'Neill.
A Morgan 40-20 led to another penalty goal before his wobbly one-pointer.