Sharp completed a move to the South Coast club in the January transfer window, reuniting with manager Nigel Adkins and assistant Crosby whom he worked with at Glanford Park.
His goals helped fire Saints to automatic promotion, and Crosby is convinced he will continue his scoring form when he runs out in the top flight for the first time.
"He has got more than enough to step up," Crosby said.
"I think he is in great physical shape and he's got more than enough quality to score goals in the Premier League."
Crosby and Adkins were first paired together as a management team at Scunthorpe in November 2006, when they replaced Brian Laws in a caretaker capacity.
They made the job their own with some impressive results, and their time in charge of the Iron saw the League One title won - and although relegation followed that, a League One play-off final win at Wembley the following season set up a season of survival in the Championship before Southampton came calling.
The Saints were struggling at the bottom of League One, and had just sacked Alan Pardew. Since then the Saints have enjoyed two straight promotions and will now mix with the biggest sides in the country. Crosby is confident the club can go on to survive in the top flight.
"Pre-season has gone OK. We're working with a great group of players at a good club and it's a season for us to look forward to," he added.
Many outside observers will feel the Saints should be aiming purely at staying in the top flight, but Adkins has far loftier aims.
“We’ve got to drive on, strive to be the best and Europe is the goal,” he said.
“We can draw inspiration from Norwich and Swansea last season in how they acquitted themselves.Southampton Football Club ultimately have got to be in the division to get into Europe and have the dream of winning the division.”
Southampton open their Premier League campaign on Sunday with a trip to face champions Manchester City.