Burns led Surrey to the county championship this season and the 28-year-old has been the leading run scorer in the top flight with 1,319 championship runs at an average of more than 69.
Kent batsman Joe Denly and Warwickshire fast bowler Olly Stone also earned their first test call-ups.
Denly has nine ODI caps but has not been in the England set-up since 2010.
Cook, England's most-capped test player with 161, retired internationally this month with an emotional century against India.
Stone was also named in the one-day squad that was announced on Wednesday as was all-rounder Ben Stokes.
Stokes and batsman Alex Hales, who is in the ODI squad, were charged this week with bringing the game into disrepute.
Both are due to appear before a disciplinary hearing in December to face charges resulting from a fracas outside a Bristol nightclub a year ago.
Stone was handed his first ODI call up as a result of Yorkshire fast bowler Liam Plunkett being ruled out of the first three fixtures due to his wedding.
Plunkett will, however, be available for the fourth and fifth ODIs.
Test squad: Joe Root (captain), Moeen Ali, Jimmy Anderson, Jonny Bairstow (wicket), Stuart Broad, Rory Burns, Jos Buttler, Sam Curran, Joe Denly, Keaton Jennings, Jack Leach, Olly Pope, Adil Rashid, Ben Stokes, Olly Stone, Chris Woakes.
(Reporting by Alan Baldwin, editing by Toby Davis)
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