Nagelsmann, whose sacking in March 2023 led to Tuchel's arrival, was tipped to return to Bayern, but has now extended his contract with Germany until 2026
It was winner takes all in Glasgow with three points meaning certain qualification for the knockout stages, and a tense match was settled by Modric's stunning curler in the 62nd minute and Perisic's flicked header late on.
Callum McGregor's powerful strike three minutes before half time had cancelled out Nikola Vlasic's opener, but Steve Clarke's Scotland were second best, and are still to reach the knockout stages of a major tournament.
Perisic's third means Croatia now finish second in Group D, and will play the runner-up of Group E in Copenhagen on June 28 in the last 16 knockout stage. England's 1-0 win over Czech Republic means they top the group, with the Czechs through as one of four best third-placed teams.