Manchester United striker Danny Welbeck scored a well-taken winner for England in the 78th minute after the introduction of substitute Theo Walcott swung a see-saw Group D battle back in his side's favour.
But an own goal from Glen Johnson on 49 minutes followed by a header from Olof Mellberg in the 59th minute threatened to set up a disastrous night for Roy Hodgson's side.
"It's always nice to come off the bench and show what you can do," said Walcott.
"We felt from first game (1-1 draw with France) we wanted to pick things up and to get three goals is terrific."
"It was a typical game of two-halves but we always had the belief. There's a great spirit about this squad."
England's first ever competitive victory over the Swedes means they will qualify for the quarter-finals with only a point in their final group game against co-hosts Ukraine in Donetsk next Tuesday.
However, Hodgson will want to see a marked improvement from his side if they are to secure the draw they need after a performance littered with errors and shaky defending.
Sweden meanwhile had to wait until 20 minutes before threatening the England goal, captain Zlatan Ibrahimovic letting fly with a long range shot that was blocked comfortably by Joe Hart.
For a while, it looked as if England's familiar failings of conceding possession too cheaply and poor delivery would be costly.
Captain Steven Gerrard had clearly seen enough on 23 minutes.
Seizing possession in midfield, the England skipper looked up to curl in a pinpoint long-range diagonal ball which the twisting Carroll met with a thunderous header that gave Isaksson no chance.
England's positive first half display came unstuck, however, within five minutes of the restart as a rejuvenated Sweden dominated play.
The equaliser came after an Ibrahimovic free-kick bounced back off the wall and was played back to an unmarked Mellberg being played onside by Johnson.
The bearded centre-half's shot beat Hart and Johnson was unable to scramble out of his own net. Mellberg claimed it, but the equaliser was ruled an own-goal by Johnson.
Milner, who had capped a poor evening by conceding the free-kick which led to Sweden's second, was then promptly hauled off for Walcott.
The Arsenal winger made an immediate impact and within minutes England were level, Walcott's shot from just outside the area catching Isaksson off-balance to make it 2-2.
Sweden continued to cause England problems on the break and Ibrahimovic forced Hart to dive at full-stretch to turn around a 76th-minute effort.
But two minutes later, Walcott rescued England again, darting into the box with an electrifying burst to cross for Welbeck, who flicked in a deft finish with the back of his heel to make it 3-2.