The financial details of each player's transfer have not been disclosed but Inter confirmed their arrivals through a series of interviews published on their official website.
Kuzmanovic scored 17 goals in 96 Bundesliga appearances before leaving Stuttgart for Italy on Thursday, while Schelotto and Kovacic arrive from Atalanta and Dinamo Zagreb respectively.
Serbian midfielder Kuzmanovic, 25, joins Inter having previously spent two and a half years with Fiorentina, an inconsistent spell that preceded a 10million euro move to Germany in 2009.
He told inter.it: "I'm delighted to have joined a great club like Inter, that the deal went through and also that I'm back in Italy.
"I still speak a bit of Italian because I learnt it in Florence. I really can't wait to get out on the pitch with my new team-mates.
"I trained with them for the first time this morning [Thursday] and I obviously need to get to know them better as players, but I'm very happy."
Inter were keen to strengthen their attacking midfield options after selling Philippe Coutinho to Liverpool in the wake of Wesley Sneijder's high-profile move to Galatasaray.
Buenos Aires-born Schelotto, 23, has swapped Serie A clubs by leaving Atalanta for Inter in an undisclosed arrangement.
The winger said: "I want to say goodbye to Atalanta, the team, the fans, the coach, everyone who always believed in me.
"But I'm glad I've joined Inter, the biggest club in the world. I hope to be able to give my very best for this team.
"I'm thrilled to be here and I've been welcomed with open arms."
Croatian playmaker Kovacic, who has drawn comparisons to Luka Modric, said: "When I was still a boy, Inter were winning lots of trophies."
The 18-year-old added: "They're one of the biggest clubs in Europe and Italy. I'm happy I've joined a team like this and I hope to stay here for a long time."