Britain had been chasing the game from the first end and went further behind when Canada scored three in the fifth end to take a 6-3 lead. But they worked hard to stay in touch and eventually gave themselves a real chance to win in the 10th end.
When she came to play her last stone of the end Muirhead was looking at a straightforward draw within the 8ft ring to score two and put the game into an extra end. However, the Scot opted to try for a triple take-out on three clustered Canadian stones that would have given her a score of three and the win. Muirhead came up short, though, as her shot removed two of the
Canadian stones but did not have the energy to displace the third, which instead rolled into shot position inside two of Muirhead's own. Muirhead acknowledged her poor start - saying her team were always on the back foot - but added: "It was always going to be a close game and we had a great chance in the last end.
"We played a couple of good ends and managed to force them into ones. It's always a big game against Canada and there are a lot of positives to take out of that one. There are also a lot of games left and we knew our first three were going to be tough."
Pressed about the choice she made at the end, Muirhead added: "I don't go for shots if I don't think they are there. I definitely thought it was there; we had a good chance."
Jennifer Jones, the Canada skip, said: "Either team could have won. In the last end we made a really good shot on my first one and it saved the game for us. We were in a little bit of trouble but were lucky enough not to have to go to the extra end."
Britain sit seventh in the rankings, two wins behind the joint leaders Canada and Switzerland. They face China in today's morning session.
Later, David Murdoch's British men's team produced a workmanlike display to beat Switzerland 4-2 and record their third victory so far.
After a cagey game that featured four blank ends in total, Britain blanked the ninth end and scored one point in the 10th with a Murdoch take-out to win.
The skip said: "That was a much better performance, we came out really firing. We were always going to stay ahead of them and we just pushed and pushed them the whole game. We dominated throughout and were in control.
"It's always tense as this is the Olympics, so you don't want to miss a shot, never mind lose a game, but we're 3-1 and I would have taken that at the start of the week".
Murdoch's team play the USA this afternoon.