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
The German youngster was played through on goal in the 42nd minute and kept his composure to evade out-rushing City keeper Ederson before rolling the ball into an empty net.
Favourites City, in their first Champions League final, started slowly and Chelsea's Timo Werner wasted one great chance early on before drawing a save from Ederson.
City, who surprisingly left experienced midfielder Fernandinho on the bench, gradually hit their stride but were unable to gain control of the match.
Phil Foden almost got through for City but a covering tackle by Antonio Rudiger denied him while Riyad Mahrez was agonisingly close to getting on the end of Kyle Walker's low cross.