England are taking five forwards to the 24-team tournament in France with Sturridge selected despite doubts over a calf problem which forced him to miss Friday's friendly win over Australia.
The 18-year-old Rashford, who signed a new long-term deal with Manchester United on Monday, became the youngest player to score on his England debut in the game against Australia at the Stadium of Light.
Delph pulled out of the squad with an injury last week and Ross Barkley was preferred to Drinkwater.
England 23-man squad for Euro 2016
Goalkeepers: Fraser Forster (Southampton), Joe Hart (Manchester City), Tom Heaton (Burnley)
Defenders: Ryan Bertrand (Southampton), Gary Cahill (Chelsea), Nathaniel Clyne (Liverpool), Danny Rose (Tottenham Hotspur), Chris Smalling (Manchester United), John Stones (Everton), Kyle Walker (Tottenham Hotspur)
Midfielders: Dele Alli (Tottenham Hotspur), Ross Barkley (Everton), Eric Dier (Tottenham Hotspur), Jordan Henderson (Liverpool), Adam Lallana (Liverpool), James Milner (Liverpool), Raheem Sterling (Manchester City), Jack Wilshere (Arsenal)
Forwards: Harry Kane (Tottenham Hotspur), Marcus Rashford (Manchester United), Wayne Rooney (Manchester United, capt), Daniel Sturridge (Liverpool), Jamie Vardy (Leicester City)
Original source: Rashford, Sturridge make England Euro squad