Fullback Leigh Halfpenny kicked three penalties and a conversion as Wales seized on errors from the visitors to claim a sixth win in succession for the first time since 2012.
Hooker Stuart McInally scored the only try for the Scots, who had further opportunities but could not break down an organised home defence.
The victory provides a solid, if unspectacular, start to Wales coach Warren Gatland’s final year in charge as he builds to the 2019 Rugby World Cup in Japan.
"It’s a start for us," Gatland told reporters. "We were a bit rusty at times. For long periods of the game we were pretty comfortable without the ball and defended well.
"We scored some nice tries and will be better for that. Scotland put us under a lot of pressure but we defended well."
Halfpenny kicked three penalties inside the opening 20 minutes as referee Mathieu Raynal from France punished Scotland for repeated indiscretions in what was a poor opening quarter.
After Scotland had replied with a penalty of their own, earned at the first scrum of the game, Wales pressed home their territorial superiority with the opening try.
The ball moved along the backline before North burst through the tackles of Scotland centres Huw Jones and Alex Dunbar, and held off another defender for a fine individual score and his 34th test try.
Scotland finished the stop-start first half strongly, though, as McInally barged over the tryline to reduce the home side’s advantage to 14-10 at the break.
Wales seized the initiative early in the second period when Davies, playing his first test in a year, burst away in the midfield and ran unopposed to the line as the Scotland defence disintegrated.
Wales were reduced to 14 players with 11 minutes to play when replacement hooker Elliot Dee was penalised for repeated infringement from the team and was sent to the sin-bin.
Peter Horne thought he had scored a second try for the Scots when he collected a kick behind the Wales defence, but the Television Match Official ruled he had knocked on in the act of dotting the ball down and Wales were able to secure the win.
"The players have worked hard in the last couple of weeks and there’s no doubt we’ve probably flattened them a little bit. They might have some heavy legs," Gatland said after a limp finish to the game from his side.
Wales handed a debut to flyhalf Jarrod Evans for the final minutes and winger Luke Morgan also played his first test, though he barely had a touch of the ball.