If Bjarsmyr's absence was to blame for the second goal, England's third owed entirely to his presence as, in attempting to clear, his swinging left foot diverted an Onuoha header from Lee Cattermole's right-wing cross into his own net.
Cattermole almost made it four on the stroke of half-time but his fierce strike from a neat Agbonlahor lay-off flew just over.
Theo Walcott's pace created a number of openings for himself throughout the second half, but each time his shooting let him down.
England paid the price for such profligacy as Sweden - galvanised by substitutes Labinot Harbuzi, Robin Soder and Molins - gradually chipped away at the deficit.
With 20 minutes remaining, Molins beat Kieran Gibbs down the right and his cross was slotted in by Berg from eight yards out.
Pearce had previously lost three semi-finals as England player and coach
Seven minutes later Onuoha fouled Toivonen 22 yards out and the Swedish midfielder picked himself up and fired the resulting free-kick past Joe Hart to make it 3-2.
And Sweden - benefitting from England's inability to retain possession - grabbed a deserved equaliser on 81 minutes when Soder flicked the ball back at the far post and Berg was on hand to crash his seventh goal of the tournament into the roof of the net.
England's task in extra-time was made all the harder when substitute Campbell picked up his second booking for a lunging foul on Mikael Lustig.
Predictably, Sweden came closest to winning the game but Berg was denied a hat-trick when his header struck the crossbar and Hart was able to tip a 25-yard shot from Harbuzi over the bar.
The record of England's senior team in penalty shoot-outs is far from impressive, while memories of the heartbreaking defeat for the Under-21s two years ago were also fresh in the mind of England fans as the game went to penalties.
Milner gave England the worst possible start, slipping and horribly miscuing his penalty high and wide.
But the hosts were unable to capitalise though as Berg's penalty was saved by Hart.
The teams then traded four successful penalties each, during which Hart was booked for straying off his line to earn a yellow card that will keep him out of the final, although Pearce has stated he intends to appeal the decision.
Gibbs slotted England's sixth penalty but Molins' effort struck the outside of the post to give England victory.
This page is best viewed in an up-to-date web browser with style sheets (CSS) enabled. While you will be able to view the content of this page in your current browser, you will not be able to get the full visual experience. Please consider upgrading your browser software or enabling style sheets (CSS) if you are able to do so.