Rashid was included in a 13-man squad for the Edgbaston game even though he has opted out of playing first-class County Championship cricket for Yorkshire this season.
Michael Vaughan, another ex-England skipper, described the decision as a "stab in the back" for the county game while Yorkshire were unhappy with the move.
Asked if Rashid, 30, may find it difficult to give his best in England's 1,000th Test, which starts on Wednesday, Cook said: "I think probably in the past he would have.
"(But) over the last 18 months I think he's matured as a cricketer and improved since we last saw him in an England Test shirt.
"I think he'll be fine. Clearly he's going to be nervous but I genuinely think he will cope -- he's bowling brilliantly."
Cook voiced his sympathy for Rashid, who has had to deal with the fallout following his selection rather than being allowed to celebrate his first call-up since the last of his 10 Test caps in December 2016.
"I can understand why it's caused a bit of fuss," he said. "But you just have to get on with it and I think we should be concentrating on the positives rather than the negatives.
"We've got a different style of English spinner with a little bit of mystery to him, who's bowling really well."
Cook does not believe a significant precedent has been set to clear the way for what would be Rashid's first Test on home soil if he makes the team on Wednesday.
"I don't think it will happen very many times again," he said. "Obviously, (national selector) Ed (Smith) said you need to be playing red-ball cricket (to be selected in future) and I think that's right.
"But sometimes in exceptional circumstances, selection goes a different way than you would like -- and obviously Ed and the selectors have made a brave call."