Cook has taken charge at Ipswich, succeeding Paul Lambert in the hot seat at Portman Road as the club hunt promotion this season.
The Englishman comes with a big reputation having done notable jobs at clubs he has previously managed.
Cook earned promotions with each of his last three clubs in the shape of Chesterfield, Portsmouth and Wigan Athletic, and now his former charge Webster expects to see the manager do the same again at Ipswich.
“He’s done a great job wherever he has gone. He knows how to get teams up I’m sure he will do well at Ipswich”, Webster, who played for Ipswich from 2016 until 2018, told Ipswich’s official site.
The squad Cook will have at his disposal at the League One club is a good one, Webster feels, and he is also sure that the way the manager plays will excite the Tractor Boys faithful.
“They have got a good squad there and he is the right appointment by the owner.
“I think he can get the club up this season.
“He has his way of playing which is exciting to watch and it’s exciting to play in.
“The supporters will enjoy it.
“It’s the right fit for sure.”
Cook was in the stands as his side registered a 2-1 victory over Accrington Stanley in their last match on Tuesday.
The win, which was the side’s third successive victory, helped the Tractor Boys move up to seventh just two points behind sixth-placed Doncaster Rovers.