Lacklustre Wales, second best for an hour under a closed roof at the Principality Stadium in Cardiff, sparked into life when Roberts barged over and North touched down after a strong diagonal run.
Wales, who recovered from 13-0 down to draw 16-16 with Ireland in their tournament opener, scored the game's opening try through scrumhalf Gareth Davies but Scotland soon replied with a try by winger Tommy Seymour.
Three Greig Laidlaw penalties put Scotland 16-13 up midway through the second half before Wales pulled clear. Duncan Taylor's late converted try proved only a consolation.
"It was a tough game, we had to dig deep and we had to grind out a victory," Wales coach Warren Gatland told reporters.
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"We had to use everything that we had in the tank to get the victory."
Scotland, beaten 15-9 at home by England a week ago, have not defeated Wales since 2007 and have not won in Cardiff for 14 years.
"We were in front until the 65th minute and we let them score so it’s there for everyone to see," lamented Scotland coach Vern Cotter.
"There are no excuses from us, we’re just going to keep working."
Wales began with a bang when scrumhalf Davies, who appeared suspiciously offside, crossed after six minutes.
Flyhalf Dan Biggar chipped into Scotland's half and when the ball was knocked back Davies picked up and ran 40 meters to the tryline.
Scotland, however, quickly hit back.
They worked downfield through numerous well-controlled phases before flyhalf Finn Russell's chip to the corner was touched down by the impressive Seymour.
Scotland were hammered 51-3 on their last visit to Cardiff in 2014 when they had Stuart Hogg sent off, and the fullback enjoyed a brighter afternoon until injury forced him off after half an hour.
Scotland continued to make meters and were rewarded with two Laidlaw penalties, either side of one from Biggar, to put the visitors 13-10 up at the break.
Further shots at goal from both sides kept Scotland ahead and a sign that the Scots were winning the battle up front came when Wales coach Warren Gatland hauled off three of his pack in one fell swoop.
Winger Tom James's electric 60-metre burst down the flank sparked the hosts into life on 58 minutes with Scotland center Taylor making a try-saving tackle on the left winger.
The crowd ignited and Wales exerted intense pressure before Roberts crashed over from close range to put his side ahead for the first time since their early try.
North made the game safe when he ran in unopposed from the Scotland 22, leaving Vern Cotter's men to wonder just where their next Six Nations win will come.
(Editing by Ed Osmond)
