Bairstow's unbeaten 108 off 48 balls, with eight fours and nine sixes, see Punjab secure the win with eight balls to spare
Stokes was ruled out of the Edgbaston Test due to a calf injury sustained in England's series levelling 330-run second Test win against the visitors at Old Trafford.
Finn 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, who failed to pick up a single wicket in the first Test defeat against Pakistan, was dropped from the second of the four Tests in the series.