Two Russell penalties and a converted George Turner try saw Gregor Townsend's side race into a 13-0 lead, before Wales hit back through Ken Owens.
Kyle Steyn crossed twice in the second half before Blair Kinghorn's superb score and Matt Fagerson's late try.
It ended their Warren Gatland hoodoo and will have fans daring to dream.
For Wales, it was a second defeat in a row after slumping to a 34-10 loss to Ireland.
Wales had their chances to score in the opening half, but failed to take advantage of all their possession and territory. How they suffered for it. This wasn't just an end to Gatland's dominance over Scotland that stretches back 11 Tests, it was an utter deconstruction, principally in a second half where Scotland attacked in devastating wave after devastating wave.
They took a while to find their ruthlessness, but once they hit their stride, guided by the mesmeric Russell, they stormed away to win and are now two wins from two for the first time in the history of the Six Nations. France to come in Paris in a fortnight. Scotland will believe that anything is possible right now.
They had a 6-0 lead early on through two Russell penalties, but for much of the opening 40 it was Wales who were in control. Behind on the scoreboard, but on top in pretty much every other sense.
They were undone by their own lack of accuracy and by Scotland's desperate scrambling. They had a lineout five metres from the home line but the towering Richie Gray pinched it. They had a scrum five metres out but their backline came up offside and the chance went.
Dan Biggar had a shot at goal. A long way out, for sure, but how many times has he nailed such kicks? He missed. They had another attacking lineout after Russell's restart went out on the full, but when they looked menacing, Jamie Ritchie pilfered it on the floor. Then, of course, Wales conceded. All those promising moments, but it was Scotland who landed the first heavy blow.
It was Turner who drove powerfully through the Wales cover off a lineout maul, a thumping finish from a hooker growing every week in stature. Russell's conversion made it 13-0, but there was drama to come directly after.
Scotland messed up at the restart, Wales attacked and Turner came steaming in a little high on George North. He got binned and in his absence, Owens piled over from close range for a score that Wales clearly deserved.
The half didn't end without another bout of painful Welsh profligacy when Rio Dyer spilled it on the left wing with the Scottish line at his mercy. They had 70% territory in that opening half and had done most of the attacking, but trailed by six at the break.
They were to pay for their wastefulness because Scotland came out with a different mindset. They survived the remaining minutes of the Turner yellow and then went after their visitors. Kinghorn, who had come on early after a Stuart Hogg head injury assessment, came into it. Duhan van der Merwe, hushed to this point, started to fire up his engines.
Turner came within a whisker of scoring his second, but Scotland didn't have to wait for long.
Piling pressure on, and forcing penalties, they put Wales where they didn't want to be. When the next chance came, Russell's half-break through the gap and his sumptuous offload close to the right wing put Steyn in. A glorious moment from the fly-half, who then walloped over the conversion from the touchline.
They turned the screw from there. More pressure and a yellow card for Liam Williams for persistent offending. They could have taken a simple three points, but gambled, went for touch on the left, then Russell cross-kicked expertly to the right and Steyn caught and scored. Suddenly, from being a close affair it was an 18-point game.
Scotland continued to unload, the beautiful execution we saw from them last week now laying waste again. Kinghorn got the bonus point try that had its beginnings in a huge forwards maul before Russell cross-kicked again with gorgeous accuracy. Van der Merwe claimed it and fed Kinghorn, who ran away to score. Thirty points now. A rout.
And there was more. Russell, utterly unplayable by now, pulled all the strings once more, flinging a precise pass over the heads of the retreating defence to the mighty Fagerson to touch down in the corner. Thirty-five points. Better than the wildest dream of the most ardent Scottish fan.
We wondered if Scotland could handle the pressure of finally backing up one win with another. We pondered if they were actually the real deal or not. The answer came, in the most emphatic style.
A stunning playmaking performance from Scotland's fly-half, who becomes the first Scotland player to notch a hat-trick of try assists in the Six Nations.
Former Scotland captain John Barclay on BBC One: "There was a maturity and a clear change of tactic second half and the ability to implement that. They kicked it long, put pressure on Wales. Same as last week, their ability to execute under pressure was brilliant.
That allows players like Finn Russell, Duhan van der Merwe grow in the game. The most pleasing thing was their ability to get into the sheds at half-time and grab that game by the scruff off the neck."
Former Wales fly-half Jonathan Davies on BBC One: "The effort was there and the youngsters stood up. They're the future for Wales.
"My problem is the lack of creativity behind and that's been a problem for a number of years.
"Today they were one-dimensional, not accurate and slightly clueless if we're being honest. That is what Wales must work on now."
Scotland: 15-Hogg; 14-Steyn, 13-Jones, 12-Tuipulotu, 11-Van der Merwe; 10-Russell, 9-White; 1-Schoeman, 2-Turner, 3-Z. Fagerson, 4-R. Gray, 5-Gilchrist, 6-Ritchie (capt), 7-Crosbie, 8-M. Fagerson
Replacements: 16-Brown, 17-Bhatti, 18-WP Nel, 19-J. Gray, 20-Dempsey, 21-Horne, 22-Kinghorn, 23-Harris
Wales: 15-L. Williams; 14-Adams, 13-North, 12-Hawkins, 11-Dyer; 10-Biggar, 9-T. Williams; 1-W. Jones, 2-Owens (capt), 3-Lewis, 4-Jenkins, 5-Beard, 6-Tshiunza, 7-Reffell, 8-Morgan
Replacements: 16-Baldwin, 17-Carre, 18-Brown, 19-Davies, 20-Faletau, 21-Webb, 22-Patchell, 23-Cuthbert