Jones crossed twice and Sean Maitland also went over as the Scots surged into a 22-6 lead at half-time and never looked back.
Scotland have not beaten the auld enemy for a decade and suffered a 61-21 drubbing at Twickenham a year ago.
Scotland were unchanged from the side that beat France 32-26 last time out, with the mercurial Russell retained in the starting XV by coach Gregor Townsend.
"Off the field we try to put different situations to players and try to discuss different scenarios", the England forwards coach said.
The fiery nature of this fixture was reflected by BBC reports of a pre-match scuffle in the tunnel between England's Owen Farrell, who went to score all his side's points and Scotland No 8 Ryan Wilson.
Under the Australian, England have won 24 out of 25 Tests, with the only blot on the record a defeat by Ireland last season that denied them a Grand Slam.
Asked if that had had any bearing on the match, Eddie Jones said: "I don't know, I don't think so, these things happen".
Ex-British and Irish Lion Shaw wants to see England set the tone with another strong start, just as they did in a tense 12-6 victory over Wales at Twickenham before the break.
Scotland had not scored a try against England at Murrayfield since wing Simon Danielli went over in 2004.
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He scored two of the Scots' three first-half tries - one an individual score from near halfway - as they blew away England before halftime then weathered an inevitable response from the visitors with a brilliant defensive effort and indomitable spirit.
Laidlaw missed the tricky conversion but Scotland led 15-6.
So often England's inspiration, Farrell provided the fast start to the second half they needed, skirting around Grant Gilchrist as the lock was caught narrow and converting his own score.
His other try might end up being the best of the tournament, breaking through a tackle just inside England's half and running 50 metres before bursting through two defenders to score.
Farrell cut the deficit to nine points early in the second half with a try the inside centre duly converted.
England came out for the second half 22-6 down - their largest halftime deficit in 136 meetings against Scotland - and scored their only try nearly immediately through second five-eighth Owen Farrell.
Farrell had a second try ruled out following a knock on in the build-up but any hopes of closing the gap further were dealt a major blow when Underhill was shown yellow for a unsafe tackle in the 66th minute.
England then found themselves a man down with 14 minutes left when Sam Underhill was sin-binned for a risky challenge on opposing replacement Jamie Bhatti.