The 31-year-old winger swapped League One for League Two when he completed the move to Fratton Park on Wednesday.
Roberts and Cook played together at Accrington before teaming up once again at the Proact Stadium.
“I’ve known him a long time and he trusts me. It’s a good fit and I was always going to try and go with him,” Roberts told BBC Radio Solent.
“He’s got the best out of me of any manager in my career, he’s had success in this division before and this club is crying out for success.”
Cook made the surprising move to the south-coast in May after guiding the Spireites to the play-off finals.
Chesterfield turned down an initial bid for the former Swindon and Ipswich player, but Pompey managed to reach an agreement to sign Roberts for an undisclosed fee.
“I’m over the moon the deal’s done and it’s been sorted out,” he added.
“It had a couple of snags along the way, but the club have come and got me and I’m glad to be here.”