England now face Argentina on Friday without the win they so badly craved from their opening fixture, while Sweden play Nigeria.
Eriksson's team made a positive, direct start to the match - pushing forward quickly to try and exploit a perceived lack of pace within the Sweden defence.
And their early endeavour was eventually rewarded with a goal.
An attacking foray by Ashley Cole led to the game's first corner - and Sol Campbell converted David Beckham's subsequent curling kick with a bullet header.
Campbell - who had a headed goal disallowed against Argentina in the 1998 World Cup - charged away to celebrate his first England goal.
Cole shot viciously from long range eight minutes before half-time, but the ball was too close to Hedman, who comfortably saved.
Sweden created several half chances in the early phase of the second-half and Marcus Allback felt he should have been awarded a penalty when he was challenged by Ferdinand after 51 minutes.
Conversely, England began the second half hesitantly and were punished when Alexandersson equalised after 59 minutes.
Mills failed to clear the ball, which was eventually bundled out to the Swedish midfielder, who found space in a central position and fired home from 20 yards.
Alexandersson's shot was close to David Seaman but swerved through the air and deceived the Arsenal goalkeeper.
Minutes later Seaman brilliantly saved from Teddy Lucic to prevent Sweden - now dominating - from taking the lead.
And after 64 minutes Seaman smothered another Lucic shot after a sweeping Swedish move had created a superb shooting opportunity.
England skipper David Beckham - ineffective through the second half - was substituted for Kieron Dyer after 63 minutes as England's grip on the game loosened.
Michael Owen shot against the side netting with 20 minutes left - but it was an isolated effort from a team struggling to remain on level terms.
Sweden pushed forward throughout the closing stages and continued to create half-chances.
Ragged England - by now totally lacking cohesion - held on for a point and have still not beaten Sweden since a 3-1 win in 1968.
England: Seaman, Mills, Ashley Cole, Ferdinand, Campbell,
Beckham (Dyer 63), Scholes, Hargreaves, Heskey,
Vassell (Joe Cole 74), Owen.
Subs Not Used: Sinclair, Fowler, Brown, Martyn, Bridge, Keown,
Southgate, Sheringham, Butt, James.
Sweden: Hedman, Mellberg, Mjallby, Jakobsson, Lucic, Linderoth,
Alexandersson, Ljungberg, Magnus Svensson (Anders Svensson 56),
Allback (Andreas Andersson 80), Larsson.
Subs Not Used: Patrik Andersson, Michael Svensson, Kihlstedt,
Antonelius, Edman, Jonson, Farnerud, Daniel Andersson,
Booked: Allback, Jakobsson.
Referee: Carlos Simon (Brazil).