After Hungary had opened up with the hammer and put a point on the board Scotland's women came roaring straight back and took a 3-1 lead in the second end.
However the Hungarians weren't about to lie down and levelled the match before Muirhead gave Scotland a 5-3 lead at half-time.
The Hungarians levelled again but Muirhead didn't panic, leading Scotland to an 8-5 victory in the tenth end.
Victory guarantees Scotland's women at least a tie-breaker but if they beat Russia on Thursday then they will automatically go into the Page 1-2 game, and that is all Muirhead is focussing on right now.
"Give them their dues, we didn't play badly but they played great out there," she said.
"Yes the scoreline was close, but we were in control for the whole game and there wasn’t much to worry about.
"We didn’t break away but we didn't need to – we were in control throughout.
"We've got a big game tomorrow, that's going to determine a lot. In these Championships you’ve just got to keep it going."
Meanwhile, in Group B, England's women ended their Swedish adventure with a 9-4 loss to Norway to finish the competition second from bottom.
There was disappointment for Scotland's men however as they slipped to a narrow 7-6 defeat to Sweden, ending their hopes of European glory for another year.
Scotland, led by skip Tom Brewster, went down 7-6 to end with a 3-5 record in Halstad, seventh in the 10-team table.
The Swedes end with a 7-1 record ahead of Czech Republic, who beat Russia 7-6 on Wednesday, Norway, who lost 6-5 to France, and Denmark, who won 8-6 over Hungary.
All have a 6-2 record.
Switzerland finish in fifth after a 7-5 win over Germany, who are second-bottom.
Russia are sixth and France eighth with Hungary topping up the table.