Mancini's side have won all 10 of their Group J matches and are assured of a seeding in the Euro 2020 draw on November 30.
Immobile started the goal-spree on eight minutes in Sicily, adding a second in the 33rd minute.
Roma midfielder Zaniolo scored his first goal for the national side on nine minutes, completing a brace on 64 minutes.
Nicolo Barella also scored in the first half.
Alessio Romagnoli, Jorginho's 75th-minute penalty, and goals from Riccardo Orsolini and Federico Chiesa concluded the hammering.
Edgar Babayan pulled a goal back for Armenia 10 minutes from time in a rout which could have been heavier with the Azzurri rattling the woodwork on multiple occasions.
This was not quite either Italy's biggest international score or their largest margin of victory.
They beat Egypt 11-3 in the bronze-medal match at the 1928 Olympics in Amsterdam and the USA 9-0 at Brentford in the first round at the 1948 Olympics. In 1920, they beat France 9-4 in a friendly.