Vern Cotter hailed his Scotland side’s second-half display, in which a haul of 20 unanswered points secured a first victory over Wales since 2007.
In round three of the Six Nations, the Welsh had led 13-9 at the break.
But two tries, and 10 points from the boot of Finn Russell after the interval, paved the way to a 29-13 win.
“We realised we were watching them play rather than playing ourselves,” Cotter said after Scotland’s second win of the championship.
“I’m very proud of that response. The boys went out and started taking the game to the Welsh team.
“We were more assertive and organised in the second half. We applied pressure and got over the line with well-scored tries.
“It means we’re still in the competition and we can get back to work on Monday and prepare for Twickenham.”
John Barclay, captaining Scotland from the back row, became only the fourth of 14 Scotland skippers in the Six Nations era to have tasted victory in his first game leading the team.
The 30-year-old, who took over from the injured Greig Laidlaw, was cautiously optimistic about Scotland’s chances against England at Twickenham on 11 March.
He told BBC Sport: “We won very well against Ireland (in round one), then we didn’t play particularly well (against France). We wanted to get out of that cycle of having a good win, then not backing it up.
“The second half, to go out there, no panicking and play with control and accuracy – we recovered from a poor first half to go on and beat a very good Welsh side.
“We believe within the group that we can do something. We go to England for the next game. We’ll have a look at them. If we play well, we can win.
“If we play like we did in Paris, if we play like we did in the first half (against Wales), then it becomes very difficult.”
England can re-take top spot in the Six Nations table from Ireland with victory over Italy on Sunday.
New Zealander Cotter has only two games remaining as Scotland head coach – the penultimate being the Calcutta Cup match – before he makes way for Gregor Townsend.
“Real guts and desire, the boys threw their bodies into it,” was Cotter’s assessment of his team’s battling performance.
“We were competitive at the breakdown so, all in all, I’m happy we came away with the win.
“We will enjoy the evening, it’s been a few years since we beat Wales. The boys can have a couple of quiet, cold beers. Then we go down to England.
“I think these experiences for the young players are great. John (Barclay) did a great job out there and steadied the ship.”