The 26-year-old completed his move to the London Stadium on Monday, signing a three-year deal with his boyhood club.
Since brekaing onto the scene as a teenager, Wilshere's career has been blighted by injuries that have prevented him form developing fully, but Pellegrini insists that the deal represents good business for West Ham.
"We are very happy to welcome Jack to West Ham United," he said. "I look forward to working with him, starting at our training camp here in Switzerland.
"Jack is a player with excellent technical ability, combined with great energy and commitment, and has proven his quality both in the Premier League and at international level.
"Most importantly, he is hungry to play for West Ham, and determined to reproduce the level of performance that he has shown in the past.
"At 26, he still has many years of football in front of him, and I believe he will be a great asset for us."
Wilshere could make his 'Hammers' debut in their pre-season friendly against Wycombe Wanderers on Saturday.