Parent company BMW announced in October last year that it was ending Mini's WRC involvement after less than three years.
It had already transferred official works status from original partner Prodrive to Motorsport Italia following a breakdown in relations with the British firm at the start of the season.
Although Prodrive is continuing to run Minis, it has not been able to put a full-year programme together so far. Jarkko Nikara will appear in a Prodrive-built car for round two in Sweden.
Former Production WRC frontrunner Kosciuszko had already been entered in a Motorsport Italia car for Monte Carlo and Sweden, and the team has now confirmed that the deal will run for the full season. Sister team Ralliart Italy had fielded Kosciuszko's PWRC Mitsubishis for the past two years.
Team boss Bruno di Pianto expects the 27-year-old Pole to make a quick transition to the WRC's top class.
"This change shouldn't be too hard, and we'll make it as easy as possible," said di Pianto.
"The cars are different and the programme is fuller, but whichever WRC series you are in the level of competition is high and the drivers who you battle for the title are in cars of the same specification as yours."
Motorsport Italia's best result of 2012 was fifth in Germany with ex-Subaru works driver Chris Atkinson, who replaced Armindo Araujo for the final rounds.