Having overcome Crystal Palace 2-0 in their previous game of the pre-season tournament, Jurgen Klopp's side looked sharp and cohesive as they scored two first-half goals following Islam Slimani's 12th-minute opener for Leicester.
Chelsea next head to France where they face Inter Milan in Nice (19:05 BST) in the International Champions Cup on Saturday before taking on Arsenal in Dublin on 1 August (20:05 BST) in the same tournament.
Pep Guardiola's side, taking part in the tournament after winning last season's Champions League, made the ideal start as Alvarez struck after 40 seconds, chesting in from close range after Nathan Ake's long-range strike hit the post.
The hosts are already through to the Germany tournament but did not take things easy as Kylian Mbappe hit a hat-trick that included a 40-yard strike. Teenager Warren Zaire-Emery was one of seven different first-half scorers.
The West Africans add to their previous Nations Cup wins in 1992 and 2015 and become the first tournament hosts to triumph since Egypt lifted the trophy in 2006.
Nigeria has struggled through issues with their federation this summer, but they're into the medal rounds of the men's Olympic soccer tournament after a 2-0 win over Denmark on Saturday.