Stokes was ruled out of the Edgbaston Test due to a calf injury sustained in England's series-levelling 330-run victory over the visitors in the second Test at Old Trafford.
Finn, who has taken 120 wickets in 33 Tests at an average of under 30 apiece, was selected ahead of Jake Ball and leg-spinner Adil Rashid, who were earlier called into the squad for the third Test.
"Finny is coming in for Ben Stokes, and (otherwise we are) as we were," Cook told reporters on Tuesday.
"Unless the pitch changes a lot from what I saw yesterday - we still have the right to change - we'll be going with the four seamers and one spinner."
Finn took six for 79 against Australia at Edgbaston in an Ashes Test victory last year and Cook added: "We've gone for a guy with a proven Test record."
Finn was dropped from the squad for the second of the four-Test series against Pakistan after failing to take a single wicket in the first Test, which England lost by 75 runs.
England team: Alastair Cook (capt), Alex Hales, Joe Root, James Vince, Gary Ballance, Jonny Bairstow (wk), Moeen Ali, Chris Woakes, Stuart Broad, Steven Finn, James Anderson.
IMAGE: Steven Finn
Photograph: Reuters
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