The 2012 grand slam champions became the first Welsh side to win five successive matches away from home in the tournament and now face a potential title-decider against England at the Millennium stadium next Saturday.
England, who host Italy at Twickenham on Sunday, are the only unbeaten side in the championship.
Flanker Sam Warburton, captain of the grand slam team, made a triumphant return after starting on the replacements' bench in the third-round win over Italy.
Warburton took over the captaincy after Ryan Jones left the field with an injury and was named man-of-the-match for a splendid display in the loose during a scrappy match dominated by Wales' forwards.
"I was so pleased to be part of the side again," he told BBC television. "We knew it was going to be immensely tough at Murrayfield. We have to be proud of the win today.
"I'll have 24 hours to relax and we can start looking foward to England on Monday."
Wales hooker Richard Hibbard scored the only try of a match dominated by the boot. Welsh fullback Leigh Halfpenny kicked seven penalties and a conversion and Scotland scrumhalf Greig Laidlaw, the hero of the home win over Ireland in the last round, kicked six.
The Welsh poured into the Scotland half after the kickoff and demonstrated their confidence in the pack when they opted to take a scrum both times after they were awarded two free kicks.
They won a penalty at the second engagement and Halfpenny converted the simplest of kicks straight in front of the posts. Laidlaw replied immediately with a much more difficult effort from the right touchline.
Against the run of play, Scotland centre Matt Scott drove his team deep into Welsh territory with a kick which bobbled out a metre from their line.
Wales took the lineout but were penalised for offside and Laidlaw converted his second penalty.
PENALTY SHOOTOUT
Halfpenny had a chance to put Wales ahead but missed two attempts in quick succession from either side of the field in the swirling wind. A third effort hit an upright.
Winger George North then burst through the defence with a powerful run which brought his side close to the Scots' line. After a series of surges, Hibbard extended a long arm to score his first try for his country.
Scotland flyhalf Duncan Weir, making his first international start, gave his side a great opportunity to score when he hacked ahead after a clever chip kick and dragged his opposite number Mike Biggar over the line to earn a five-metres scrum.
However, Scotland were penalised by Craig Joubert to the displeasure of the crowd, whose mood quickly changed when Laidlaw kicked a fourth penalty to give his side the lead.
With the whistle about to blow for halftime, Scotland lock Jim Hamilton conceded a clumsy penalty in front of his posts when he charged into Wales scrumhalf Mike Phillips while the ball was still in a ruck. Halfpenny kicked the penalty to put his side 13-12 ahead at the interval.
Laidlaw was short with an early penalty attempt at the limit of his range at the start of the second half while Halfpenny made no mistake with a much easier chance.
Both kickers landed further penalties for the sides as the game entered the final quarter, with Halfpenny then putting his side 10 points clear with two further penalties.
Scotland, who lost lock Richie Gray in the first half with a hamstring injury, beseiged the Welsh line in the final minutes but the defence held firm and Warburton won a vital penalty which enabled Wales to clear and stay 10 points ahead.
