Leicester defender Maguire headed in the opening goal from a corner on 30 minutes, with Alli adding a second just before the hour.
Viktor Claesson drilled over for Sweden early on with an ambitious attempt from distance, while Harry Kane, the tournament's leading scorer with six goals, dragged wide after a penetrating run by Raheem Sterling.
The lively Sterling should have doubled the lead on the stroke of half-time when he brilliantly controlled Jordan Henderson's lofted pass, but he was unable to find a way past Robin Olsen and his follow-up was blocked by Andreas Granqvist.
Pickford had relatively little to do in a quiet first half, but the England goalkeeper produced a vital stop to claw away a Marcus Berg header minutes after the restart.
England promptly cranked up the pressure and pulled further ahead on 59 minutes when a completely unmarked Alli sent a thumping header past Olsen after a first-time cross by Jesse Lingard.
Maguire then lashed over after a scene of panic in the Sweden area as England chased a third goal, although they were twice indebted to Pickford for preserving their two-goal lead.
The Everton shot-stopper thrust out a hand to superbly keep out Claesson's low strike from 10 yards following a neat lay-off by Berg.
Pickford then showed terrific reflexes to tip over an effort from Berg, who shot on the turn after being picked out in the area by substitute John Guidetti.
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