Both teams struggled to hold on to the ball in an error-ridden affair, with the only points coming from the boots of Hodge and Bosch.
Melbourne conceded a stunning 18 penalties to seven, which cruelled their chances of successive victories.
Rebels hooker James Hanson also spent some extra time on the pine after being yellow-carded for collapsing a maul just two minutes after coming on in the second half.
Second in their conference, the Sharks hadn't lost at home in seven games and were rated huge favourites to continue that record.
But they were rattled early after centre Andre Esterhuizen was sent off in the 16th minute for a dangerous tackle on Melbourne's Sefa Naivalu, dumping the Fijian-born Rebel on his head after the whistle.
That setback came after they lost in-form winger Kobus van Wyk in the warm-up.
Sharks captain Tera Mtembu strangely elected to turn down several shots at goal to chase tries, with Bosch off-target with two attempts.
While Stirzaker said leaving Durban with any points was a good thing, he felt the game was there to be won.
"We had the opportunity to win it, we just couldn't string too many phases together," the halfback said.
"When we did we looked quite good in the first half and showed we could split them but credit to the Sharks, they were a tough side to grind down."