Fabio Capello became the first manager to guide England to victory over their jinx team since Alf Ramsey in 1968.
The 22nd-minute goal by the Swedish defender was also England's 2,000th since they began playing internationals in 1872.
The unlucky Majstorovic deflected a header from Gareth Barry past goalkeeper Andreas Isaksson following a left-wing cross from Stewart Downing.
"We've shown in the last couple of games we can compete with the very best," England captain John Terry told ITV Sport. "If we can continue this momentum for the next six months hopefully things will be a lot better."
Capello made eight changes to the team that beat world champions Spain 1-0 in another friendly on Saturday.
Man of the match Kyle Walker, Jack Rodwell and Bobby Zamora made their first starts for England and Daniel Sturridge came on as a 58th-minute substitute to win his first cap.
Despite their international inexperience, the home side dominated for long periods and could easily have won by a bigger margin.
Sweden would have had no complaints if they had been 3-0 down at the interval.
Erik Hamren's men, who also lost 2-0 to Denmark in a friendly on Friday, again looked lethargic and the closest they came to scoring was when Johan Elmander headed wide from a Kim Kallstrom corner after 33 minutes.
They did little else to trouble the England defence with captain Zlatan Ibrahimovic anonymous in a withdrawn role up front and it was no surprise when he was replaced at half-time.
England, after a relatively slow start, were much more dangerous with Walker and Theo Walcott combining well on the right and Leighton Baines and Downing doing likewise on the left.
Holding midfielder Phil Jones enabled Barry and Rodwell to push forward, a tactic that led to the goal.
Jones, winning his third cap, was unlucky not to score his first goal after 40 minutes when he pounced on a loose ball just inside the Sweden half and ran 40 metres before slicing his shot narrowly wide of Isaksson's left-hand post.
RODWELL MISS
Four minutes later Rodwell was guilty of a worse miss when his header hit the outside of the same post when it looked easier to score.
"It was good to beat Sweden for the first time in 43 years but it was also important for me to see Jones, Walker and Rodwell play so well," said Capello. "I was looking for something new in these two games and I have found it.
"Walker, Rodwell, Jones are good technically, physically and are very fast - and that is what is important in modern football."
Sweden came more into the game in the later stages, much as Spain did on Saturday, but they failed to find the goal that would have preserved their long unbeaten run against England.
Christian Wilhelmsson had an excellent chance in the dying minutes but somehow blasted his effort high over the bar.
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