Gareth Southgate's side went 1-0 down after Damsgaard scored for Denmark in the 30th minute.
Saka helped England equalise with a low cross into the six-yard box, which resulted in an own goal by Kjaer.
No goals were scored in the second half and the match went into extra time.
Raheem Sterling was tripped in the box by Maehle and England awarded a penalty, with Harry Kane making it 2-1 in the 103rd minute.
England will now play Italy in the Euros final on Sunday, July 11.
Photos in Leeds taken at the Left Bank Community Centre, in Cardigan Road, Burley, by Guzelian Photography.
Wembley photos taken by the Press Association.
No goals were scored in the second half and the match went into extra time.