The Storm left AAMI Park with the competition points after a 30-24 victory, but the gutsy Panthers deserved plenty of applause for pushing the home side to the final whistle.
Already missing injured Test fullback Dylan Edwards, Penrith lost Cleary to concussion after just seven minutes when he came off second best against Storm No.7 Jahrome Hughes when the pair collided off the ball.
With Jarome Luai now at Wests Tigers, that left 21-year-old five-eighth Jack Cole to call the shots in just his 13th NRL match, but the youngster rose to the challenge.
Blaize Talagi came off the bench in his club debut to replace Cleary and also held his own.
Melbourne also had casualties, with centre Nick Meaney, suffering a suspected fractured jaw after a collision with teammate Nelson Asofa-Solomona, while winger Grant Anderson limped off with a knee injury.
Despite a second straight loss, Panthers coach Ivan Cleary said his team "couldn't have tried any harder".
"I'm proud of the boys tonight," he said, adding that his son Nathan had recovered well.
"There's lots of things that we sort of didn't help ourselves throughout the game, there's clearly a lack of cohesion at times, which is understandable.
"But I thought from a cultural point of view, I thought they were very brave.
"I thought the boys couldn't have tried any harder."
The Storm lived up to their pre-match favouritism early, with tries to Anderson and second-rower Eliesa Katoa as well as a penalty setting up a 14-0 lead after 20 minutes.
But with scrappers such as Liam Henry and Liam Martin digging in, Penrith clawed their way back, with centre Paul Alamoti scoring twice, with Cole having a hand in both.
Penrith hit the front 16-14 four minutes after halftime when Casey McLean scored the easiest of tries off a 20-metre scrum.
However, the lead only lasted two minutes before Xavier Coates leapt high above the pink pack to put the Storm back in front, with Tyran Wishart then extending it to 24-16.
In typical Panthers fashion, they refused to be beaten, with winger Brian To'o making it a four-point margin.
Papenhuyzen crossed in the 72nd minute, before To'o bagged his second in the 75th minute to set up a grand-stand finish, but the Storm defence held firm.
Storm coach Craig Bellamy felt his team could have become complacent in the shock absence of Cleary.
"We had that good start, and then he goes off, and some of the players probably thought ... in the back of your mind you find a voice is telling you, 'Oh, yeah, it's going to be OK today,'" Bellamy said.
"But they (Penrith) are a really professional team, and they know how to win, and you just can't do that with them.
"It doesn't matter who they've got on their side."
Five-eighth Cameron Munster was put on report for an alleged hip-drop tackle on Martin, and Papenhuyzen also injured his ankle late in the match.
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