Both sides were decimated by State of Origin duties and injury.
When regular No.1 Ryan Papenhuyzen was a late withdrawal with calf tightness Meaney switched from centre to the custodian position he has played previously with distinction.
The Cowboys, equally well served by their own fullback Scott Drinkwater, who was their best, took an 18-12 lead in the 50th minute before Meaney stepped up with two late tries.
He also landed five from five with the boot and ran for well over 260m in a man-of-the-match display to take the Storm to second position on the ladder.
Melbourne back-rowers Shawn Blore and Eliesa Katoa played key roles in ensuring the hosts finished the match three points outside the top eight.
Drinkwater set up two tries and scored one against his former club.
The Storm's first try was pure class by Katoa who leapt above fullback Drinkwater to snare a Tyran Wishart bomb and wrangle his way over to score. The athletic forward has proven to be a master of the art in 2025 and this effort was one of the best this season.
North Queensland had one chance in the opening quarter of the match and Drinkwater set up winger Robert Derby with a sensational long ball.
Melbourne wrecking ball Nelson Asofa-Solomona came off the bench with devastating impact and crashed over from close range before Cowboys speed demon Jaxon Purdue raced 40m with pure blistering speed on the cusp of halftime.
Drinkwater gave the home side the lead after the break when he scored after a wonder ball from back-rower John Bateman and dart by half Jake Clifford.
Meaney did it all himself from 30m out with a scything run and then ran on to a Blore pass to make his personal haul for the match 18 points.
"Obviously I have played a lot (at fullback) in my career and it feels natural," Meaney said post-match.
"With the Origin boys out, we all had to step up."
Half Jahrome Hughes, who had nice touches, said the side's fifth win in a row was vital.
"It was a gritty win against a tough opponent. There was a lot of energy from our boys," he said.
"Our back-rowers have been huge for us, a big credit to them.
"A big thing for us is to be consistent and we need to do that for the rest of the year."
Cowboys coach Todd Payten said there was "a whole heap of effort" from his side but Drinkwater added that the fact the team got so close "probably hurt more" than being flogged.
"We put ourselves in a position to win," he said.
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