They didn't get across the tryline until the 61st minute, relying on the boot of fullback Jason Woodward kicking penalties.
That try came after skipper Scott Higginbotham took the ball from the back of their scrum and charged over.
The Melbourne scrum earned bragging rights over their rivals while lock Luke Jones continued to push his claims for a Wallaby call-up, making a game-high 16 runs.
The Force, missing halfback Alby Mathewson (ankle), lock Sam Wykes (rib) and winger Luke Morahan (hamstring), also struggled to bag a five-pointer with five-eighth Sias Ebersohn keeping them in the game.
They hammered hard early in the second half, getting within centimetres of the tryline but the home side's defence held up and they ended up taking three points for a ruck penalty.
The Force looked out of steam on the back of their winning run, which included the scalp of the NSW Waratahs last round.
Higginbotham's converted try extended their lead to 19-6 and the early workload started to take its toll on both sides with the error toll mounting.
Woodward brought his personal toll to 17 points - five penalties and a conversion - before Force winger Nick Cummins crossed after the siren to ensure an all-important bonus point.
Higginbotham said the win was overdue after a run of close losses, including to the champion Chiefs last round.
He praised the scrum for laying the platform; all the more impressive after losing starting prop Laurie Weeks 16 minutes in with concussion.
"The scrum was fantastic and we really built on that," Higginbotham said.
"The boys had a tough time in New Zealand so to come back here and really put it on, it was good for them.
"It was time for us to get a win."
Melbourne coach Tony McGahan was also delighted his team had finally been rewarded.
"They've trained really well and have had a tremendous attitude and, against a good Force side, we got the fruits of the last three weeks," he said.
He said he wasn't concerned about his team's inability to break away from the Force in the first half, feeling that the workload would count against the visitors as the clock wound down.
Force coach Michael Foley saw it the same way and said his team failed to match the desperate Rebels at the breakdown and scrum, particularly in the opening half.
"We weren't able to get enough momentum in the early stages of the game and that took its toll in the back stages," he said.
"We had to do a lot of work than what we should have to make up for the things we weren't doing proactively," he said.
He said his team, who host the Bulls next round, weren't looking for excuses such as the mental and physical drain of their winning run.
"There might be mitigating factors but we have to overlook those and know we could have done better."
- AAP
