Cook admitted he was awake in the middle of the night deliberating over which of the two should complete England’s seam attack against Pakistan but, in the end, Finn’s proven Test record made his mind up.
The 27-year-old will therefore replace injured all-rounder Ben Stokes, at a venue where he performed well in last summer’s Ashes success.
Cook said: “Finny is coming in for Ben Stokes, and [otherwise we’re] as we were.”
That means England are planning to pick Moeen Ali as their only spinner, with Adil Rashid again missing out.
“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,” added Cook.
The choice between Finn and Ball was more marginal.
Both sat out last week’s series-levelling victory at Old Trafford, only for Stokes to succumb to a calf injury there.
“It was a tough call – as I found out at two o’clock this morning when I was wide awake thinking about it,” said Cook.
“We’ve gone for a guy who’s got a proven Test record, who’s taken 120 wickets and has a knack of taking wickets.
“He has good memories here of last year. He hasn’t quite been at his best this year - there have been a few factors in that, but physically he’s now better and has had a couple of things sorted out in the last week.”
Pakistan, meanwhile are playing a cagey game as they finalise plans for the pivotal Edgbaston Test.
The tourists appear likely to drop out-of-form opener Shan Masood, in favour of the inexperienced specialist Sami Aslam or a move up the order for No 3 Azhar Ali.
Either way, captain Misbah-ul-Haq was in no mood to let England in on their selection choices. He did make one thing clear, however – that Pakistan, who need to put last week’s poor performance in Manchester behind them quickly with the Investec series level at 1-1, will not be victims of any muddled thinking.
“We are sure about who will be playing – absolutely no doubt but I don’t want to let the other team know,” said Misbah.
“Keep guessing! We are clear in our thinking, and we know what we are going to do.”
Should 20-year-old Aslam get the nod – despite just one innings of 17 on tour, against Worcestershire last weekend – Misbah insists he will be ready.
“I think anyone who is here in the Pakistan squad is capable of that,” added the captain.
“We expect everybody to deliver.”
He expects too that the stern words, spoken by coach Mickey Arthur after Pakistan’s 330-run defeat at Old Trafford, will be heeded.
“His anger is appropriate,” said Misbah. “The players are also unhappy that, after an outstanding performance, you were nowhere in the next game. But you have to give credit to the opposition, they played so outstandingly that it put you out of the match.”