Prop Ollie Norris brushed off tackles and scored for the Chiefs in the 80th minute to deliver heartbreak for the Rebels at AAMI Park.
Melbourne were looking to rebound from a 43-point thrashing by the Blues and while they didn't get the win they did plenty to earn back respect.
Their loss meant a clean sweep of Australian teams by their Kiwi rivals through round 13.
Chiefs skipper Sam Cane said his finals-bound team "got out of jail".
"We knew that the Rebels were a better side than what they showed on the points table and they weren't going to be the same side that turned up to Eden Park last week," Cane told Stan Sport.
"They showed that for the full 80 minutes."
Rebels skipper Michael Wells said it was "incredibly disappointing".
"You don't get rewarded for participation so that's where we sit," he said.
"It's a big step forward ... but we're still coming up a little bit short."
Melbourne took the lead in the 71st minute with Reece Hodge slotting a penalty to put his team ahead 27-26.
They extended the margin with another penalty in the 77th but it ultimately wasn't enough.
Chiefs five-eighth Josh Ioane was hero and villain for his team, setting up three of their tries but also gifting the Rebels two.
Melbourne led at the 47 minute mark after scoring two tries while they were a man down, with prop Pone Fa'amausili given a yellow card for some off the ball antics following a massive hit on Chiefs lock Naitoa Ah Kuoi.
Ioane threw a loose pass that was toed ahead by Rebels winger Andrew Kellaway for a try.
And just before halftime Rebels five-eighth Carter Gordon also intercepted an Ioane pass and dashed 80 metres to touch down.
But the Chiefs again nudged ahead 19-17 in the 57th minute when hooker Samisoni Taukei'aho drove across the line, with Ioane adding the extras.
It looked like the men from Hamilton might run away with the match after centre Emoni Narawa danced through some traffic to open up a nine point lead.
The Rebels rallied but the late try meant they still came up short.
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