The game see-sawed between the two teams five times.
The Scots started well, scoring two in the first end with a simple draw by Muirhead.
However, Sweden responded with their own two in the second end, thanks to a nice double take-out and stay by Viktorsson.
Scotland could only take a single in the third and when, after blanking the fourth, Sweden scored another two in the fifth, they had a half-time 4-3 lead.
They built on this with a single steal in the sixth and then Scotland scored two and stole a single in the eighth to take the lead again, at 6-5, when a Swedish clear-out attempt left one Scottish stone counting.
In the ninth, Muirhead could only move one Swedish stone rather than the two she was looking for, and this gave Sweden a draw for another two and a 7-6 lead.
The tenth end came down to Muirhead's last stone attempt to shift all three of a Swedish cluster, but she either got it wrong, or the angle was not there, and the Swedes stole another, for victory.
Muirhead said: "We thought that last shot was there.
"We wouldn't have gone for it if we didn't think it was there.·
"I still think it's there now and I think if my stone had swung another inch when we look at it now, we wouldn't have been far away.
"When those shots come off – great, but I missed it."
Defeat was a disappointment for 20-year-old Muirhead, who had finished second at the World Championships earlier this year.
She said: "Silver again, it's getting to be a bit of a trend.·
"These silvers will make the gold even more valuable when it comes."
Viktorsson said, "it feels fantastic to be European champion, I can't realise it yet.
"We really played as a team and we had a lot of fun out there.
"I think we did quite well, taking it easy, playing smart and playing our shots."
Earlier, Switzerland landed a bronze double when Mirjam Ott led their women's team to a 9-5 win over Russia's Ludmila Privivkova, and Christof Schwaller was a 7-4 winner over Germany's Andy Kapp.
December 2010: Muirhead eyes victory as curling championships approach climax
December 2010: Scots put their curlers in for big night out
December 2010: Curling stars aim to sweep up at European Championships