Last season, Tedesco took charge of second-tier FC Erzgebirge Aue for the last 11 games - his first job as a boss - and guided them to survival.
Tedesco has signed a two-year deal at Schalke, who finished 10th in the German top-flight.
The division's youngest coach Julian Nagelsmann, 29, has extended his deal with Hoffenheim until 2021.
Nagelsmann took over in February 2016 without a proper senior licence, but led them to a club-best fourth-placed finish, qualifying for the Champions League play-offs.
His excellent showing has since seen him linked several bigger clubs, including champions Bayern Munich.
Bundesliga side Schalke have appointed 31-year-old Domenico Tedesco as coach to replace the sacked Markus Weinzierl.