They were set on their way when Maguire rose imperiously to head home from a corner and Alli added the killer blow after the break as England showed they are not entirely dependent on the tournament's leading scorer Harry Kane for goals.
Goalkeeper Jordan Pickford then produced a series of crucial saves in the final half-hour to seal England's long-awaited spot in the last four.
England coach Gareth Southgate named an unchanged side for Saturday's World Cup quarter-final against Sweden following their penalty shootout victory over Colombia in the last round. Emil Krafth replaced the suspended Mikael Lustig in defence, while Sebastian Larsson returned from suspension in place of Gustav Svensson.
England forward Raheem Sterling has yet to score at the World Cup from his unusual position as the support striker to Kane, and his finishing let him down against Sweden.
Former Thai Navy SEAL dies preparing for cave rescue mission
The trapped team has been receiving basic training in breathing through diving equipment in case the order to evacuate is given. Authorities now say there are concerns about falling oxygen levels in the chamber where the boys and their coach are trapped.
England's Dele Alli, left, celebrates with England's Harry Maguire, center and England's Marcus Rashford, right, after scoring his side's second goal during the quarterfinal match between Sweden and England.
The surprising run is being fuelled by goals from set pieces, and there was another against Sweden when Maguire headed in a corner in the 30th minute.
Maguire outjumped the much smaller Emil Forsberg to nod low into the corner, as England scored for the eighth time in Russian Federation from a dead-ball situation.
Sweden slowed the tempo early on as the game got off to a boring start, but the first half-chance fell to England as Harry Kane flashed wide from 20 yards after good running from Raheem Sterling.
The importance of the keeper's contribution was swiftly underlined with England doubling their lead in the 59th minute as Jesse Lingard's teasing cross into the box was met by Alli unmarked at the far post to head powerfully past Olsen.