Kiev - England remained on course for a place in the knockout rounds of Euro 2012 after coming from behind to send Sweden crashing out of the tournament with a 3-2 victory on Friday.
However, the substitution of James Milner for Walcott changed the game. The Arsenal winger lashed in a 64th-minute equaliser to make it 2-2 and then provided the run and pass which set up Welbeck's winner.
"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."
Welbeck was delighted more for the team than himself.
"It was a typical game of two-halves but we always had the belief. There's a great spirit about this squad."
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.
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.
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.
England were rocking and it got worse for Hodgson's men on 59 minutes as Sweden went 2-1 up. Sebastian Larsson swung in a curling free-kick and shambolic defending in the England defence allowed Mellberg to nod home Sweden's second.