Finn, who has taken 120 wickets in 33 Tests at an average of under 30 apiece, is set to replace the injured Ben Stokes.
All-rounder Stokes has been ruled out after damaging his calf in England's 330-run win in the second Test at Old Trafford that levelled the four-match series at 1-1.
Cook said there was still a slim chance England could give a home debut to leg-spinner Adil Rashid if there was a marked change in pitch conditions between Tuesday and Wednesday's toss at Edgbaston.
England also had the option of selecting Nottinghamshire quick Jake Ball, who made his Test debut in Pakistan's 75-run win in the series opener at Lord's.
"It was a tough call as I found when I lie awoke at 2:00 am (0100 GMT) thinking about it" Cook told reporters at Edgbaston on Tuesday.
Finn took six for 79 against Australia at Edgbaston in an Ashes Test win last year and Cook added: "We've gone for a guy with a proven Test record."
England team: Alastair Cook (C), Alex Hales, Joe Root, James Vince, Gary Ballance, Jonny Bairstow (wkt), Moeen Ali, Chris Woakes, Stuart Broad, Steven Finn, James Anderson. (AFP) PDS PDS