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 after just seven minutes when he came off second best against Storm No.7 Jahrome Hughes when the pair collided off the ball.
There was a scramble of bodies after Melbourne fullback Ryan Papenhuyzen put a kick through and Cleary jumped backwards into Hughes before landing heavily, with a category-one concussion ending his night.
With Jarome Luai now at Wests Tigers, that left 21-year-old Jack Cole to call the shots in just his 13th NRL match, but the youngster rose to the challenge.
Melbourne also had casualties in 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.
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 like 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, but then To'o bagged his second in the 75th minute to set up a grand-stand finish, but the Storm defence held firm.
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