Immobile and Zaniolo were among those promoted to the side after Friday's 3-0 win in Bosnia and they delivered in style against their lowly opponents.
Almost immediately after the re-start, the home side doubled their lead when Immobile played in Zaniolo, who beat Armenian keeper Aram Hayrapetyan to net his first senior international goal.
Chiesa and Immobile both came close for the Italians but Armenia should have reduced the deficit through Aleksandre Karapetyan, who pounced on a poor back-pass but hit the bar.
Immobile might have wrapped up the first period with a hat-trick when his shot clipped the post in first-half injury time, while Chiesa also hit the woodwork after the break.
Zaniolo grabbed his second just past the hour mark and Alessio Romagnoli deflected home Italy's sixth goal of the night with 72 minutes on the clock.
Babayan fired a brilliant consolation for Armenia from outside the box in the 79th minute but Chiesa finally secured his first senior goal two minutes later when he headed home from close range.