Malan, born in London but brought up in South Africa, gets his chance after impressing for England's second-string Lions team.
Fast bowler Mills now only plays limited overs cricket because of a back condition, while Hampshire left-arm spinner Dawson is also a useful batsman.
England will be looking to build on the form that saw them finish runners-up to the West Indies in this year's World Twenty20 in India.
None of the uncapped trio, however, were included in England's 14-man squad for the five one-days internationals against Sri Lanka that will take place before the lone Twenty20 international.
Bairstow, also England's man of the series in a 2-0 campaign win over Sri Lanka, was one of three glovemen in the one-day squad along with regular white-ball keeper Jos Buttler and Kent's Sam Billings.
Middlesex batsman Eoin Morgan retained the captaincy of both the ODI and T20 squads, with Test skipper Alastair Cook not having played limited overs cricket for England since before last year's World Cup in Australia and New Zealand.
Meanwhile Test new-ball duo James Anderson and Stuart Broad were again spared white-ball duty ahead of England's four-match series in the five-day format at home to Pakistan starting next month.
"The six matches coming up against Sri Lanka will give the players a good test of their credentials.
"The T20 squad has a balanced feel to it with an exciting blend of youth and experience."
England ODI squad:
Eoin Morgan (Middlesex, capt), Moeen Ali (Worcestershire), Jonny Bairstow (Yorkshire, wkt), Jos Buttler (Lancashire, wkt), Steven Finn (Middlesex), Chris Jordan (Sussex), Alex Hales Hales (Nottinghamshire), Liam Plunkett (Yorkshire), Adil Rashid (Yorkshire), Joe Root (Yorkshire), Jason Roy (Surrey), James Vince (Hampshire), David Willey (Yorkshire), Chris Woakes (Warwickshire)
Eoin Morgan (Middlesex, capt), Jonny Bairstow (Yorkshire, wkt), Sam Billings (Kent, wkt), Jos Buttler (Lancashire, wkt), Liam Dawson (Hampshire), Chris Jordan (Sussex), Dawid Malan (Middlesex), Tymal Mills (Sussex), Liam Plunkett (Yorkshire), Adil Rashid (Yorkshire), Jason Roy (Surrey), James Vince (Hampshire), David Willey (Yorkshire)
Fixtures
Jun 21: 1st ODI, Trent Bridge
Jun 24: 2nd ODI, Edgbaston
Jun 26: 3rd ODI, Bristol
Jun 29: 4th ODI, The Oval
Jul 02: 5th ODI, Cardiff
Jul 05: T20, Southampton.
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