Scotland twice led in the first half with a try by Huw Jones and a spectacular 60-metre effort by Van der Merwe but England hit back with two for recalled winger Max Malins then edged 13-12 ahead at halftime with a penalty by Owen Farrell.
Reuters reports tries for Ellis Genge and Ben White and a penalty apiece kept it as a one-point game until Van der Merwe finished off a lovely move three minutes from the end.
After decades of suffering against England this was Scotland's third successive victory, their best run in the game's oldest fixture for more than 50 years - and means they have won four and drawn one of the last five Calcutta Cup meetings.
"After the first half, we knew we had more in us and came out for the second half with a bang and managed to get the win so we are buzzing," said Van der Merwe.
"They scored quite early and we knew we had to stay in the fight and we got some points on the board and responded. As a winger you don't get a lot of opportunities so I needed to take them.
"I was quite surprised how I got my first but I will take that one all day long. We spoke about getting our first win of the Six Nations and next week we need to take some confidence from that."
It was a game England could and should have won as they gradually turned the screw in the second half but they struggled with the final pass against ferocious Scottish defence and, with flyhalf Finn Russell pulling the strings, they always looked a threat in attack.
The game took a while to get going but, after 15 minutes of tame kicking, Scotland burst into life as Sione Tuipulotu sent a grubber through a static England defence for Jones to pounce on for the opening try.
Marcus Smith then sent a crossfield kick into the corner for Malins to cleverly touch down at full stretch for his first try in his 15th appearance.
Twickenham was then treated to a brilliant individual try by Van der Merwe, not that most of those present will have appreciated it.
The winger, who scored the only try when the Scots triumphed at Twickenham for the first time in 38 years last time they visited in 2021, caught a loose kick just inside his own half then set off on a curving, side-stepping run past five flailing defenders, whose questionable efforts would have left new defence coach Kevin Sinfield exasperated.
England hit back when finally putting the ball sharply through the hands and Malins was sent over in the corner. Farrell missed his second successive conversion but then slotted a penalty to send the hosts in 13-12 ahead.
Borthwick's first halftime team talk evidently instructed the team to hold on to the ball and it paid off with some powerful runs by the forwards that ended with prop Genge charging over eight minutes after the restart.
Scotland hit back quickly with a sniping try by surprise scrumhalf selection White.
A penalty at each end kept the crowd tense and England were doing most of the attacking, but it was the Scots who had the final, decisive word as pinpoint passing and a muscular finish by Van der Merwe snatched the thrilling victory.
England return to Twickenham next play Italy, while Scotland host Wales, who were well beaten by Ireland in Cardiff.
Scotland wing Duhan van der Merwe. Photo: Photosport/INPHO