Aidy Boothroyd’s side fell behind to a Martin Chrien header in the first half.
Alfie Mawson scrambled in an equaliser before a powerful Nathan Redmond strike gave England victory.
The Three Lions are top of Group A on four points and will guarantee a place in the last four by beating hosts Poland in Thursday’s final group game.
Only the winners of each of the three groups go through to the semi-finals, along with the best runner-up.
A 90th-minute Poland equaliser against Sweden in the other group game on Monday simplified the situation for England. If Sweden had held on for victory they would be level on points and goal difference ahead of Thursday’s decisive fixtures.