In a match in which neither side's defence impressed, England were forced to recover from trailing in the second half after Olof Mellberg forced Glen Johnson to put past his own keeper and then scored himself to cancel out Andy Carroll's headed opening goal for England on 23 minutes.
Substitute Theo Walcott brought England level when he drove a shot straight down the middle of the goal on 64 minutes from outside of the area that somehow evaded Swedish goalkeeper Andreas Isaksson as he dived to his left. It was his first international goal since his hat-trick in a 4-1 win in Croatia almost four years ago.
Welbeck won it for England on 78 minutes with both sides pushing for the three points when he backheeled Walcott's cross into the net with the opposing defenders again caught marking fresh air.
England were marginally the better side on an evening of some mediocrity with Sweden heading home from the finals after their 2-1 loss to Ukraine in the opening match was followed by this defeat. They will not be a great miss to the tournament.
England conclude their Group D campaign against Ukraine in Donetsk on Tuesday with the co-hosts needing three points to deprive England of progress after their 2-0 loss to France earlier in the day.
France and England top the section on four points with the French warm favourites to progress as group winners if they can come up with a win over the already doomed Swedes.
Roy Hodgson's side will need to improve considerably to remain a threat in this tournament, mainly in defence and in their toiling efforts at ball retention, but will be boosted by the return of the suspended Wayne Rooney to face Ukraine.
Welbeck's winning goal gave them a first success over Sweden in competitive matches after two losses and five draws against the Scandinavian country on similar occasions.
Hodgson's decision to bring back Liverpool forward Carroll to replace the Arsenal teenager Alex Oxlade-Chamberlain seemed to be vindicated when Carroll rose to power a header into the net from 12 yards out after Steven Gerrard had provided a similar delivery that enabled Joleon Lescott to score with a header in the 1-1 draw against France.
England were leading 1-0 and comfortable with Isaksson diving to swat away a Scott Parker shot and Welbeck heading wide from James Milner's cross before Carroll finally unearthed the opening goal with his first start and first goal at a major finals.
With Sweden appearing impotent from open play, Ashley Young drove a shot into the side of the net as England disappeared up the tunnel looking relatively content.
There was a significant mood swing at the outset of the second half when the unmarked and bearded Mellberg prodded the ball over the line to bring Sweden level with England's defence in a ghastly state.
Goalkeeper Joe Hart palmed the ball into the midriff of Johnson, who failed to scramble the ball clear as it dropped over the line.
Suddenly England were being tested as Johnson was forced to make a fine tackle on Rasmus Elm before Sweden moved 2-1 clear as Mellberg rose unmarked to beat Hart with a header from Sebastian Larsson's free-kick.
It was awful defending with John Terry and Lescott nowhere to be seen.
Johnson's pace was again called upon when he raced back to prevent the moody Zlatan Ibrahimovic from getting a free shot at goal as Sweden pushed for a 3-1 lead.
England looked groggy, but dragged themselves back to 2-2 when Walcott's hopeful drive evaded Isaksson seconds after the goalkeeper superbly pawed away John Terry's header inside the Sweden box.
With one goal likely to settle matters either way, Ibrahimovic forced Hart to plunge to his right to divert a ferocious shot on 77 minutes.
It was left to Welbeck to drive the dagger into the hopes of over 20,000 Sweden fans in the 70,000 crowd when he brilliantly converted from Walcott's cross to seal the win.
England could have added to their total in added time, but Isaksson came up with another eye-catching save to halt Gerrard's volley with Walcott playing provider on the break. But England's work had been done for the night.