Muirhead defeated Swedish rival Margaretha Sigfridsson in a repeat of the World Championship final earlier this year, winning by exactly the same 6-5 scoreline.
But it was close, Swedish fourth Maria Prytz had the final stone advantage heading into the decisive end but Muirhead's rink got their tactics spot on to steal it.
The victory means they head into their final two group games secure of a place in the 1-2 page play-off, which will give them two chances to reach the gold medal match this weekend.
And they will face the only other unbeaten team Germany in that match, who they also play in Thursday's final round robin session.
"A win is a win but it was a tough game and we managed to grind it out," said Muirhead, whose rink face also bottom ranked Norway on Wednesday.
"It's always a high quality game against Sweden and I'm not surprised it came down to the last stone but we kept the pressure on."
- Sports & Recreation
- Margaretha Sigfridsson
- Maria Prytz